Wireless energy transfer

ABSTRACT

Disclosed is an apparatus for use in wireless energy transfer, which includes a first resonator structure configured to transfer energy non-radiatively with a second resonator structure over a distance greater than a characteristic size of the second resonator structure. The non-radiative energy transfer is mediated by a coupling of a resonant field evanescent tail of the first resonator structure and a resonant field evanescent tail of the second resonator structure.

CROSS REFERENCE TO RELATED APPLICATIONS

Pursuant to U.S.C. § 120, this application is a continuation-in-part ofU.S. Utility patent application Ser. No. 11/481,077, filed Jul. 5, 2006,which claims priority to U.S. Provisional Application Ser. No.60/698,442, filed Jul. 12, 2005; pursuant to U.S.C. § 119(e), thisapplication claims priority to U.S. Provisional Application Ser. No.60/908,383, filed Mar. 27, 2007, and U.S. Provisional Application Ser.No. 60/908,666, filed Mar. 28, 2007; and pursuant to U.S.C. § 120, andU.S.C. § 363, this application is also a continuation-in-part ofInternational Application No. PCT/US2007/070892, filed Jun. 11, 2007.The contents of the prior applications are incorporated herein byreference in their entirety.

BACKGROUND

The disclosure relates to wireless energy transfer. Wireless energytransfer may for example, be useful in such applications as providingpower to autonomous electrical or electronic devices.

Radiative modes of omni-directional antennas (which work very well forinformation transfer) are not suitable for such energy transfer, becausea vast majority of energy is wasted into free space. Directed radiationmodes, using lasers or highly-directional antennas, can be efficientlyused for energy transfer, even for long distances (transfer distanceL_(TRANS)>>L_(DEV), where L_(DEV) is the characteristic size of thedevice and/or the source), but require existence of an uninterruptibleline-of-sight and a complicated tracking system in the case of mobileobjects. Some transfer schemes rely on induction, but are typicallyrestricted to very close-range (L_(TRANS)<<L_(DEV)) or low power (˜mW)energy transfers.

The rapid development of autonomous electronics of recent years (e.g.laptops, cell-phones, house-hold robots, that all typically rely onchemical energy storage) has led to an increased need for wirelessenergy transfer.

SUMMARY

The inventors have realized that resonant objects with coupled resonantmodes having localized evanescent field patterns may be used fornon-radiative wireless energy transfer. Resonant objects tend to couple,while interacting weakly with other off-resonant environmental objects.Typically, using the techniques described below, as the couplingincreases, so does the transfer efficiency. In some embodiments, usingthe below techniques, the energy-transfer rate can be larger than theenergy-loss rate. Accordingly, efficient wireless energy-exchange can beachieved between the resonant objects, while suffering only modesttransfer and dissipation of energy into other off-resonant objects. Thenearly-omnidirectional but stationary (non-lossy) nature of the nearfield makes this mechanism suitable for mobile wireless receivers.Various embodiments therefore have a variety of possible applicationsincluding for example, placing a source (e.g. one connected to the wiredelectricity network) on the ceiling of a factory room, while devices(robots, vehicles, computers, or similar) are roaming freely within theroom. Other applications include power supplies for electric-enginebuses and/or hybrid cars and medical implantable devices.

In some embodiments, resonant modes are so-called magnetic resonances,for which most of the energy surrounding the resonant objects is storedin the magnetic field; i.e. there is very little electric field outsideof the resonant objects. Since most everyday materials (includinganimals, plants and humans) are non-magnetic, their interaction withmagnetic fields is minimal. This is important both for safety and alsoto reduce interaction with the extraneous environmental objects.

In one aspect, an apparatus is disclosed for use in wireless energytransfer, which includes a first resonator structure configured totransfer energy with a second resonator structure over a distance Dgreater than a characteristic size L₂ of the second resonator structure.In some embodiments, D is also greater than one or more of: acharacteristic size L₁ of the first resonator structure, acharacteristic thickness T₁ of the first resonator structure, and acharacteristic width W₁ of the first resonator structure. The energytransfer is mediated by evanescent-tail coupling of a resonant field ofthe first resonator structure and a resonant field of the secondresonator structure. The apparatus may include any of the followingfeatures alone or in combination.

In some embodiments, the first resonator structure is configured totransfer energy to the second resonator structure. In some embodiments,the first resonator structure is configured to receive energy from thesecond resonator structure. In some embodiments, the apparatus includesthe second resonator structure.

In some embodiments, the first resonator structure has a resonantangular frequency ω₁, a Q-factor Q₁, and a resonance width Γ₁, thesecond resonator structure has a resonant angular frequency ω₂, aQ-factor Q₂, and a resonance width Γ₂, and the energy transfer has arate κ. In some embodiments, the absolute value of the difference of theangular frequencies ω₁ and ω₂ is smaller than the broader of theresonant widths Γ₁ and Γ₂.

In some embodiments Q₁>100 and Q₂>100, Q₁>300 and Q₂>300, Q₁>500 andQ₂>500, or Q₁>1000 and Q₂>1000. In some embodiments, Q₁>100 or Q₂>100,Q₁>300 or Q₂>300, Q₁>500 or Q₂>500, or Q₁>1000 or Q₂>1000.

In some embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 0.5},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 1},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 2},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 5.$

In some such embodiments, D/L₂ may be as large as 2, as large as 3, aslarge as 5, as large as 7, or as large as 10.

In some embodiments, Q₁>1000 and Q₂>11000, and the coupling to lossratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 10},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 25},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 40.$

In some such embodiments, D/L₂ may be as large as 2, as large as 3, aslarge as 5, as large as 7, as large as 10.

In some embodiments, Q_(κ)=ω/2κ is less than about 50, less than about200, less than about 500, or less than about 1000. In some suchembodiments, D/L₂ is as large as 2, as large as 3, as large as 5, aslarge as 7, or as large as 10.

In some embodiments, the quantity κ/√{square root over (Γ₁Γ₂)} ismaximized at an angular frequency {tilde over (ω)} with a frequencywidth {tilde over (Γ)} around the maximum, and

the absolute value of the difference of the angular frequencies ω₁ and{tilde over (ω)} is smaller than the width F, and

the absolute value of the difference of the angular frequencies ω₂ and{tilde over (ω)} is smaller than the width {tilde over (Γ)}.

In some embodiments, the energy transfer operates with an efficiencyη_(work) greater than about 1%, greater than about 10%, greater thanabout 30%, greater than about 50%, or greater than about 80%.

In some embodiments, the energy transfer operates with a radiation lossη_(rad) less that about 10%. In some such embodiments the coupling toloss ratio

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq {0.1.}$

In some embodiments, the energy transfer operates with a radiation lossη_(rad) less that about 1%. In some such embodiments, the coupling toloss ratio

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 1.$

In some embodiments, in the presence of a human at distance of more than3 cm from the surface of either resonant object, the energy transferoperates with a loss to the human η_(k) of less than about 1%. In somesuch embodiments the coupling to loss ratio

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 1.$

In some embodiments, in the presence of a human at distance of more than10 cm from the surface of either resonant object, the energy transferoperates with a loss to the human η_(k) of less than about 0.2%. In somesuch embodiments the coupling to loss ratio

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 1.$

In some embodiments, during operation, a device coupled to the first orsecond resonator structure with a coupling rate Γ_(work) receives ausable power P_(work) from the resonator structure.

In some embodiments, P_(work) is greater than about 0.01 Watt, greaterthan about 0.1 Watt, greater than about 1 Watt, or greater than about 10Watt.

In some embodiments, if the device is coupled to the first resonator,then 1/2≦[(Γ_(supply)/Γ₁)²−1]/(κ/√{square root over (Γ₁Γ₂)})²≦2, or1/4≦[(Γ_(supply)/Γ₁)²−1]/(κ/√{square root over (Γ₁Γ₂)})²≦4, or1/8≦[(Γ_(supply)/Γ₁)²−1]/(κ/√{square root over (Γ₁Γ₂)})²≦8and, if thedevice is coupled to the second resonator, then1/2≦[(Γ_(supply)/Γ₁)²−1]/(κ/√{square root over (Γ₁Γ₂)})²≦2, or1/4≦[(Γ_(supply)/Γ₁)²−1]/(κ/√{square root over (Γ₁Γ₂)})²≦4, or1/8≦[(Γ_(supply)/Γ₁)²−1]/(κ/√{square root over (Γ₁Γ₂)})²≦8

In some embodiments, the device includes an electrical or electronicdevice. In some embodiments, the device includes a robot (e.g. aconventional robot or a nano-robot). In some embodiments, the deviceincludes a mobile electronic device (e.g. a telephone, or a cell-phone,or a computer, or a laptop computer, or a personal digital assistant(PDA)). In some embodiments, the device includes an electronic devicethat receives information wirelessly (e.g. a wireless keyboard, or awireless mouse, or a wireless computer screen, or a wireless televisionscreen). In some embodiments, the device includes a medical deviceconfigured to be implanted in a patient (e.g. an artificial organ, orimplant configured to deliver medicine). In some embodiments, the deviceincludes a sensor. In some embodiments, the device includes a vehicle(e.g. a transportation vehicle, or an autonomous vehicle).

In some embodiments, the apparatus further includes the device.

In some embodiments, during operation, a power supply coupled to thefirst or second resonator structure with a coupling rate Γ_(supply)drives the resonator structure at a frequency f and supplies powerP_(total). In some embodiments, the absolute value of the difference ofthe angular frequencies ω=2πf and ω₁ is smaller than the resonant widthΓ₁, and the absolute value of the difference of the angular frequenciesω=2πf and ω₂ is smaller than the resonant width Γ₂. In some embodiments,f is about the optimum efficiency frequency.

In some embodiments, if the power supply is coupled to the firstresonator, then 1/2≦[(Γ_(supply)/Γ₁)²−1]/(κ/√{square root over(Γ₁Γ₂)})²≦2, or 1/4≦[(Γ_(supply)/Γ₁)²−1]/(κ/√{square root over(Γ₁Γ₂)})²≦4, or 1/8≦[(Γ_(supply)/Γ₁)²−1]/(κ/√{square root over(Γ₁Γ₂)})²≦8, and, if the power supply is coupled to the secondresonator, then 1/2≦[(Γ_(supply)/Γ₂)²−1]/(κ/√{square root over(Γ₁Γ₂)})²≦2, or 1/4≦[(Γ_(supply)/Γ₂)²−1]/(κ/√{square root over(Γ₁Γ₂)})²≦4, or 1/8≦[(Γ_(supply)/Γ₂)²−1]/(κ/√{square root over(Γ₁Γ₂)})²≦8.

In some embodiments, the apparatus further includes the power source.

In some embodiments, the resonant fields are electromagnetic. In someembodiments, f is about 50 GHz or less, about 1 GHz or less, about 100MHz or less, about 10 MHz or less, about 1 MHz or less, about 100 KHz orless, or about 10 kHz or less. In some embodiments, f is about 50 GHz orgreater, about 1 GHz or greater, about 100 MHz or greater, about 10 MHzor greater, about 1 MHz or greater, about 100 kHz or greater, or about10 kHz or greater. In some embodiments, f is within one of the frequencybands specially assigned for industrial, scientific and medical (ISM)equipment.

In some embodiments, the resonant fields are primarily magnetic in thearea outside of the resonant objects. In some such embodiments, theratio of the average electric field energy to average magnetic filedenergy at a distance D_(p) from the closest resonant object is less than0.01, or less than 0.1. In some embodiments, L_(R) is the characteristicsize of the closest resonant object and D_(p)/L_(R) is less than 1.5, 3,5, 7, or 10.

In some embodiments, the resonant fields are acoustic. In someembodiments, one or more of the resonant fields include a whisperinggallery mode of one of the resonant structures.

In some embodiments, one of the first and second resonator structuresincludes a self resonant coil of conducting wire, conducting Litz wire,or conducting ribbon. In some embodiments, both of the first and secondresonator structures include self resonant coils of conducting wire,conducting Litz wire, or conducting ribbon. In some embodiments, both ofthe first and second resonator structures include self resonant coils ofconducting wire or conducting Litz wire or conducting ribbon, and Q₁>300and Q₂>300.

In some embodiments, one or more of the self resonant conductive wirecoils include a wire of length l and cross section radius a wound into ahelical coil of radius r, height h and number of turns N. In someembodiments, N=√{square root over (l²−h²)}/2πr.

In some embodiments, for each resonant structure r is about 30 cm, h isabout 20 cm, a is about 3 mm and N is about 5.25, and, during operation,a power source coupled to the first or second resonator structure drivesthe resonator structure at a frequency f. In some embodiments, f isabout 10.6 MHz. In some such embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 40},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 15},$

or

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 5},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 1.$

In some such embodiments D/L_(R) is as large as about 2, 3, 5, or 8.

In some embodiments, for each resonant structure r is about 30 cm, h isabout 20 cm, a is about 1 cm and N is about 4, and, during operation, apower source coupled to the first or second resonator structure drivesthe resonator structure at a frequency f. In some embodiments, f isabout 13.4 MHz. In some such embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 70},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 19},$

or

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 8},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 3.$

In some such embodiments D/L_(R) is as large as about 3, 5, 7, or 10.

In some embodiments, for each resonant structure r is about 10 cm, h isabout 3 cm, a is about 2 mm and N is about 6, and, during operation, apower source coupled to the first or second resonator structure drivesthe resonator structure at a frequency f. In some embodiments, f isabout 21.4 MHz. In some such embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 59},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 15},$

or

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 6},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 2.$

In some such embodiments D/L_(R) is as large as about 3, 5, 7, or 10.

In some embodiments, one of the first and second resonator structuresincludes a capacitively loaded loop or coil of conducting wire,conducting Litz wire, or conducting ribbon. In some embodiments, both ofthe first and second resonator structures include capacitively loadedloops or coils of conducting wire, conducting Litz wire, or conductingribbon. In some embodiments, both of the first and second resonatorstructures include capacitively loaded loops or coils of conducting wireor conducting Litz wire or conducting ribbon, and Q₁>300 and Q₂>300.

In some embodiments, the characteristic size L_(R) of the resonatorstructure receiving energy from the other resonator structure is lessthan about 1 cm and the width of the conducting wire or Litz wire orribbon of said object is less than about 1 mm, and, during operation, apower source coupled to the first or second resonator structure drivesthe resonator structure at a frequency f. In some embodiments, f isabout 380 MHz. In some such embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 14.9},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 3.2},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 1.2},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq {0.4.}$

In some such embodiments, D/L_(R) is as large as about 3, about 5, about7, or about 10.

In some embodiments, the characteristic size of the resonator structurereceiving energy from the other resonator structure L_(R) is less thanabout 10 cm and the width of the conducting wire or Litz wire or ribbonof said object is less than about 1 cm, and, during operation, a powersource coupled to the first or second resonator structure drives theresonator structure at a frequency f. In some embodiments, f is about 43MHz. In some such embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 15.9},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 4.3},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 1.8},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq {0.7.}$

In some such embodiments, D/L_(R) is as large as about 3, about 5, about7, or about 10.

In some embodiments, the characteristic size L_(R) of the resonatorstructure receiving energy from the other resonator structure is lessthan about 30 cm and the width of the conducting wire or Litz wire orribbon of said object is less than about 5 cm, and, during operation, apower source coupled to the first or second resonator structure drivesthe resonator structure at a frequency f. In some such embodiments, f isabout 9 MHz. In some such embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 67.4},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 17.8},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 7.1},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq {2.7.}$

In some such embodiments, D/L_(R) is as large as about 3, about 5, about7, or about 10.

In some embodiments, the characteristic size of the resonator structurereceiving energy from the other resonator structure L_(R) is less thanabout 30 cm and the width of the conducting wire or Litz wire or ribbonof said object is less than about 5 mm, and, during operation, a powersource coupled to the first or second resonator structure drives theresonator structure at a frequency f. In some embodiments, f is about 17MHz. In some such embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 6.3},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 1.3},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq {0.5.}},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq {0.2.}$

In some such embodiments, D/L_(R) is as large as about 3, about 5, about7, or about 10.

In some embodiments, the characteristic size L_(R) of the resonatorstructure receiving energy from the other resonator structure is lessthan about 1 m, and the width of the conducting wire or Litz wire orribbon of said object is less than about 1 cm, and, during operation, apower source coupled to the first or second resonator structure drivesthe resonator structure at a frequency f. In some embodiments, f isabout 5 MHz. In some such embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 6.8},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 1.4},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 0.5},$

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq {0.2.}$

In some such embodiments, D/L_(R) is as large as about 3, about 5, about7, or about 10.

In some embodiments, during operation, one of the resonator structuresreceives a usable power P_(w) from the other resonator structure, anelectrical current I_(s) flows in the resonator structure which istransferring energy to the other resonant structure, and the ratio

$\frac{I_{s}}{\sqrt{P_{w}}}$

is less than about 5 Amps/√{square root over (Watts)} or less than about2 Amps/√{square root over (Watts)}. In some embodiments, duringoperation, one of the resonator structures receives a usable power P_(w)from the other resonator structure, a voltage difference V_(s) appearsacross the capacitive element of the first resonator structure, and theratio

$\frac{V_{s}}{\sqrt{P_{w}}}$

is less than about 2000 Volts/√{square root over (Watts)} or less thanabout 4000 Volts/√{square root over (Watts)}.

In some embodiments, one of the first and second resonator structuresincludes a inductively loaded rod of conducting wire or conducting Litzwire or conducting ribbon. In some embodiments, both of the first andsecond resonator structures include inductively loaded rods ofconducting wire or conducting Litz wire or conducting ribbon. In someembodiments, both of the first and second resonator structures includeinductively loaded rods of conducting wire or conducting Litz wire orconducting ribbon, and Q₁>300 and Q₂>300.

In some embodiments, the characteristic size of the resonator structurereceiving energy from the other resonator structure L_(R) is less thanabout 10 cm and the width of the conducting wire or Litz wire or ribbonof said object is less than about 1 cm, and, during operation, a powersource coupled to the first or second resonator structure drives theresonator structure at a frequency f. In some embodiments, f is about 14MHz. In some such embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 32},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 5.8},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 2},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq {0.6.}$

In some such embodiments, D/L_(R) is as large as about 3, about 5, about7, or about 10.

In some embodiments, the characteristic size L_(R) of the resonatorstructure receiving energy from the other resonator structure is lessthan about 30 cm and the width of the conducting wire or Litz wire orribbon of said object is less than about 5 cm, and, during operation, apower source coupled to the first or second resonator structure drivesthe resonator structure at a frequency f. In some such embodiments, f isabout 2.5 MHz. In some such embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 105},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 19},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 6.6},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq {2.2.}$

In some such embodiments, D/L_(R) is as large as about 3, about 5, about7, or about 10.

In some embodiments, one of the first and second resonator structuresincludes a dielectric disk. In some embodiments, both of the first andsecond resonator structures include dielectric disks. In someembodiments, both of the first and second resonator structures includedielectric disks, and Q₁>300 and Q₂>300.

In some embodiments, the characteristic size of the resonator structurereceiving energy from the other resonator structure is L_(R) and thereal part of the permittivity of said resonator structure ∈ is less thanabout 150. In some such embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 42.4},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 6.5},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 2.3},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq {0.5.}}$

In some such embodiments, D/L_(R) is as large as about 3, about 5, about7, or about 10.

In some embodiments, the characteristic size of the resonator structurereceiving energy from the other resonator structure is L_(R) and thereal part of the permittivity of said resonator structure ∈ is less thanabout 65. In some such embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 30.9},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq 2.3},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} \geq {0.5.}$

In some such embodiments, D/L_(R) is as large as about 3, about 5, about7.

In some embodiments, at least one of the first and second resonatorstructures includes one of: a dielectric material, a metallic material,a metallodielectric object, a plasmonic material, a plasmonodielectricobject, a superconducting material.

In some embodiments, at least one of the resonators has a quality factorgreater than about 5000, or greater than about 10000.

In some embodiments, the apparatus also includes a third resonatorstructure configured to transfer energy with one or more of the firstand second resonator structures,

where the energy transfer between the third resonator structure and theone or more of the first and second resonator structures is mediated byevanescent-tail coupling of the resonant field of the one or more of thefirst and second resonator structures and a resonant field of the thirdresonator structure.

In some embodiments, the third resonator structure is configured totransfer energy to one or more of the first and second resonatorstructures.

In some embodiments, the first resonator structure is configured toreceive energy from one or more of the first and second resonatorstructures.

In some embodiments, the first resonator structure is configured toreceive energy from one of the first and second resonator structures andtransfer energy to the other one of the first and second resonatorstructures.

Some embodiments include a mechanism for, during operation, maintainingthe resonant frequency of one or more of the resonant objects. In someembodiments, the feedback mechanism comprises an oscillator with a fixedfrequency and is configured to adjust the resonant frequency of the oneor more resonant objects to be about equal to the fixed frequency. Insome embodiments, the feedback mechanism is configured to monitor anefficiency of the energy transfer, and adjust the resonant frequency ofthe one or more resonant objects to maximize the efficiency.

In another aspect, a method of wireless energy transfer is disclosed,which method includes providing a first resonator structure andtransferring energy with a second resonator structure over a distance Dgreater than a characteristic size L₂ of the second resonator structure.In some embodiments, D is also greater than one or more of: acharacteristic size L₁ of the first resonator structure, acharacteristic thickness T₁ of the first resonator structure, and acharacteristic width W₁ of the first resonator structure. The energytransfer is mediated by evanescent-tail coupling of a resonant field ofthe first resonator structure and a resonant field of the secondresonator structure.

In some embodiments, the first resonator structure is configured totransfer energy to the second resonator structure. In some embodiments,the first resonator structure is configured to receive energy from thesecond resonator structure.

In some embodiments, the first resonator structure has a resonantangular frequency ω₁, a Q-factor Q₁, and a resonance width F₁, thesecond resonator structure has a resonant angular frequency ω₂, aQ-factor Q₂, and a resonance width Γ₂, and the energy transfer has arate κ. In some embodiments, the absolute value of the difference of theangular frequencies ω₁ and ω₂ is smaller than the broader of theresonant widths Γ₁ and Γ₂.

In some embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 0.5},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 1},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 2},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 5.$

In some such embodiments, D/L₂ may be as large as 2, as large as 3, aslarge as 5, as large as 7, or as large as 10.

In another aspect, an apparatus is disclosed for use in wirelessinformation transfer which includes a first resonator structureconfigured to transfer information by transferring energy with a secondresonator structure over a distance D greater than a characteristic sizeL₂ of the second resonator structure. In some embodiments, D is alsogreater than one or more of: a characteristic size L₁ of the firstresonator structure, a characteristic thickness T₁ of the firstresonator structure, and a characteristic width W₁ of the firstresonator structure. The energy transfer is mediated by evanescent-tailcoupling of a resonant field of the first resonator structure and aresonant field of the second resonator structure.

In some embodiments, the first resonator structure is configured totransfer energy to the second resonator structure. In some embodiments,the first resonator structure is configured to receive energy from thesecond resonator structure. In some embodiments the apparatus includes,the second resonator structure.

In some embodiments, the first resonator structure has a resonantangular frequency ω₁, a Q-factor Q₁, and a resonance width Γ₁, thesecond resonator structure has a resonant angular frequency ω₂, aQ-factor Q₂, and a resonance width Γ₂, and the energy transfer has arate κ. In some embodiments, the absolute value of the difference of theangular frequencies ω₁ and ω₂ is smaller than the broader of theresonant widths Γ₁ and Γ₂.

In some embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 0.5},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 1},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 2},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 5.$

In some such embodiments, D/L₂ may be as large as 2, as large as 3, aslarge as 5, as large as 7, or as large as 10.

In another aspect, a method of wireless information transfer isdisclosed, which method includes providing a first resonator structureand transferring information by transferring energy with a secondresonator structure over a distance D greater than a characteristic sizeL₂ of the second resonator structure. In some embodiments, D is alsogreater than one or more of: a characteristic size L₁ of the firstresonator structure, a characteristic thickness T₁ of the firstresonator structure, and a characteristic width W₁ of the firstresonator structure. The energy transfer is mediated by evanescent-tailcoupling of a resonant field of the first resonator structure and aresonant field of the second resonator structure.

In some embodiments, the first resonator structure is configured totransfer energy to the second resonator structure. In some embodiments,the first resonator structure is configured to receive energy from thesecond resonator structure.

In some embodiments, the first resonator structure has a resonantangular frequency ω₁, a Q-factor Q₁, and a resonance width Γ₁, thesecond resonator structure has a resonant angular frequency ω₂, aQ-factor Q₂, and a resonance width Γ₂, and the energy transfer has arate κ. In some embodiments, the absolute value of the difference of theangular frequencies ω₁ and ω₂ is smaller than the broader of theresonant widths Γ₁ and Γ₂.

In some embodiments, the coupling to loss ratio

${\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 0.5},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 1},{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 2},$

or

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} > 5.$

In some such embodiments, D/L₂ may be as large as 2, as large as 3, aslarge as 5, as large as 7, or as large as 10.

It is to be understood that the characteristic size of an object isequal to the radius of the smallest sphere which can fit around theentire object. The characteristic thickness of an object is, when placedon a flat surface in any arbitrary configuration, the smallest possibleheight of the highest point of the object above a flat surface. Thecharacteristic width of an object is the radius of the smallest possiblecircle that the object can pass through while traveling in a straightline. For example, the characteristic width of a cylindrical object isthe radius of the cylinder.

The distance D over which the energy transfer between two resonantobjects occurs is the distance between the respective centers of thesmallest spheres which can fit around the entirety of each object.However, when considering the distance between a human and a resonantobject, the distance is to be measured from the outer surface of thehuman to the outer surface of the sphere.

As described in detail below, non-radiative energy transfer refers toenergy transfer effected primarily through the localized near field,and, at most, secondarily through the radiative portion of the field.

It is to be understood that an evanescent tail of a resonant object isthe decaying non-radiative portion of a resonant field localized at theobject. The decay may take any functional form including, for example,an exponential decay or power law decay.

The optimum efficiency frequency of a wireless energy transfer system isthe frequency at which the figure of merit

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}}$

is maximized, all other factors held constant.

The resonant width (Γ) refers to the width of an object's resonance dueto object's intrinsic losses (e.g. loss to absorption, radiation, etc.).

It is to be understood that a Q-factor (Q) is a factor that compares thetime constant for decay of an oscillating system's amplitude to itsoscillation period. For a given resonator mode with angular frequency tand resonant width Γ, the Q-factor Q=ω/2Γ.

The energy transfer rate (κ) refers to the rate of energy transfer fromone resonator to another. In the coupled mode description describedbelow it is the coupling constant between the resonators.

It is to be understood that Q_(κ)=ω/2κ.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention belongs. In case of conflict withpublications, patent applications, patents, and other referencesmentioned incorporated herein by reference, the present specification,including definitions, will control.

Various embodiments may include any of the above features, alone or incombination. Other features, objects, and advantages of the disclosurewill be apparent from the following detailed description.

Other features, objects, and advantages of the disclosure will beapparent from the following detailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic of a wireless energy transfer scheme.

FIG. 2 shows an example of a self-resonant conducting-wire coil.

FIG. 3 shows a wireless energy transfer scheme featuring twoself-resonant conducting-wire coils

FIG. 4 shows an example of a capacitively loaded conducting-wire coil,and illustrates the surrounding field.

FIG. 5 shows a wireless energy transfer scheme featuring twocapacitively loaded conducting-wire coils, and illustrates thesurrounding field.

FIG. 6 shows an example of a resonant dielectric disk, and illustratesthe surrounding field.

FIG. 7 shows a wireless energy transfer scheme featuring two resonantdielectric disks, and illustrates the surrounding field.

FIGS. 8 a and 8 b show schematics for frequency control mechanisms.

FIGS. 9 a through 9 c illustrate a wireless energy transfer scheme inthe presence of various extraneous objects.

FIG. 10 illustrates a circuit model for wireless energy transfer.

FIG. 11 illustrates the efficiency of a wireless energy transfer scheme.

FIG. 12 illustrates parametric dependences of a wireless energy transferscheme.

FIG. 13 plots the parametric dependences of a wireless energy transferscheme.

FIG. 14 is a schematic of an experimental system demonstrating wirelessenergy transfer.

FIGS. 15-17. Plot experiment results for the system shown schematicallyin FIG. 14.

DETAILED DESCRIPTION

FIG. 1 shows a schematic that generally describes one embodiment of theinvention, in which energy is transferred wirelessly between tworesonant objects.

Referring to FIG. 1, energy is transferred, over a distance D, between aresonant source object having a characteristic size L₁ and a resonantdevice object of characteristic size L₂. Both objects are resonantobjects. The source object is connected to a power supply (not shown),and the device object is connected to a power consuming device (e.g. aload resistor, not shown). Energy is provided by the power supply to thesource object, transferred wirelessly and non-radiatively from thesource object to the device object, and consumed by the power consumingdevice. The wireless non-radiative energy transfer is performed usingthe field (e.g. the electromagnetic field or acoustic field) of thesystem of two resonant objects. For simplicity, in the following we willassume that field is the electromagnetic field.

It is to be understood that while two resonant objects are shown in theembodiment of FIG. 1, and in many of the examples below, otherembodiments may feature 3 or more resonant objects. For example, in someembodiments a single source object can transfer energy to multipledevice objects. In some embodiments energy may be transferred from afirst device to a second, and then from the second device to the third,and so forth.

Initially, we present a theoretical framework for understandingnon-radiative wireless energy transfer. Note however that it is to beunderstood that the scope of the invention is not bound by theory.

Coupled Mode Theory

An appropriate analytical framework for modeling the resonantenergy-exchange between two resonant objects 1 and 2 is that of“coupled-mode theory” (CMT). The field of the system of two resonantobjects 1 and 2 is approximated by F(r,t)≈a₁(t)F₁(r)+a₂(t)F₂(r), whereF_(1,2)(r) are the eigenmodes of 1 and 2 alone, normalized to unityenergy, and the field amplitudes a_(1,2)(t) are defined so that|a_(1,2)(t)|² is equal to the energy stored inside the objects 1 and 2respectively. Then, the field amplitudes can be shown to satisfy, tolowest order:

$\begin{matrix}{{\frac{a_{1}}{t} = {{{- {\left( {\omega_{1} - {\; \Gamma_{1}}} \right)}}a_{1}} + {\; \kappa \; a_{2}}}}{{\frac{a_{2}}{t} = {{{- {\left( {\omega_{2} - {\; \Gamma_{2}}} \right)}}a_{2}} + {\; \kappa \; a_{1}}}},}} & (1)\end{matrix}$

where ω_(1,2) are the individual angular eigenfrequencies of theeigenmodes, Γ_(1,2) are the resonance widths due to the objects'intrinsic (absorption, radiation etc.) losses, and K is the couplingcoefficient. Eqs. (1) show that at exact resonance (ω₁=ω₂ and Γ₁=Γ₂),the eigenmodes of the combined system are split by 2κ; the energyexchange between the two objects takes place in time ˜π/2κ and is nearlyperfect, apart for losses, which are minimal when the coupling rate ismuch faster than all loss rates (κ>>Γ_(1,2)). The coupling to loss ratioκ/√{square root over (Γ₁Γ₂)} serves as a figure-of-merit in evaluating asystem used for wireless energy-transfer, along with the distance overwhich this ratio can be achieved. The regime κ/√{square root over(Γ₁Γ₂)}>>1 is called “strong-coupling” regime.

In some embodiments, the energy-transfer application preferably usesresonant modes of high Q=ω/2Γ, corresponding to low (i.e. slow)intrinsic-loss rates Γ. This condition may be satisfied where thecoupling is implemented using, not the lossy radiative far-field, butthe evanescent (non-lossy) stationary near-field.

To implement an energy-transfer scheme, usually finite objects, namelyones that are topologically surrounded everywhere by air, are moreappropriate. Unfortunately, objects of finite extent cannot supportelectromagnetic states that are exponentially decaying in all directionsin air, since, from Maxwell's Equations in free space: {right arrow over(k)}²=ω²/c² where {right arrow over (k)} is the wave vector, ω theangular frequency, and c the speed of light. Because of this, one canshow that they cannot support states of infinite Q. However, verylong-lived (so-called “high-Q”) states can be found, whose tails displaythe needed exponential or exponential-like decay away from the resonantobject over long enough distances before they turn oscillatory(radiative). The limiting surface, where this change in the fieldbehavior happens, is called the “radiation caustic”, and, for thewireless energy-transfer scheme to be based on the near field ratherthan the far/radiation field, the distance between the coupled objectsmust be such that one lies within the radiation caustic of the other.Furthermore, in some embodiments, small Q_(κ)=ω/2κ corresponding tostrong (i.e. fast) coupling rate κ is preferred over distances largerthan the characteristic sizes of the objects. Therefore, since theextent of the near-field into the area surrounding a finite-sizedresonant object is set typically by the wavelength, in some embodiments,this mid-range non-radiative coupling can be achieved using resonantobjects of subwavelength size, and thus significantly longer evanescentfield-tails. As will be seen in examples later on, such subwavelengthresonances can often be accompanied with a high Q, so this willtypically be the appropriate choice for the possibly-mobile resonantdevice-object. Note, though, that in some embodiments, the resonantsource-object will be immobile and thus less restricted in its allowedgeometry and size, which can be therefore chosen large enough that thenear-field extent is not limited by the wavelength. Objects of nearlyinfinite extent, such as dielectric waveguides, can support guided modeswhose evanescent tails are decaying exponentially in the direction awayfrom the object, slowly if tuned close to cutoff, and can have nearlyinfinite Q.

In the following, we describe several examples of systems suitable forenergy transfer of the type described above. We will demonstrate how tocompute the CMT parameters ω_(1,2), Q_(1,2) and Q_(κ) described aboveand how to choose these parameters for particular embodiments in orderto produce a desirable figure-of-merit κ/√{square root over(Γ₁Γ₂)}=√{square root over (Q₁Q₂)}/Q_(κ). In particular, this figure ofmerit is typically maximized when ω_(1,2) are tuned to a particularangular frequency {tilde over (ω)}, thus, if {tilde over (Γ)} is halfthe angular-frequency width for which √{square root over (Q₁Q₂)}/Q_(κ)is above half its maximum value at {tilde over (ω)}, the angulareigenfrequencies ω_(1,2) should typically be tuned to be close to {tildeover (ω)} to within the width {tilde over (Γ)}.

In addition, as described below, Q_(1,2) can sometimes be limited notfrom intrinsic loss mechanisms but from external perturbations. In thosecases, producing a desirable figure-of-merit translates to reducingQ_(κ) (i.e. increasing the coupling). Accordingly we will demonstratehow, for particular embodiments, to reduce Q_(κ).

Self-Resonant Conducting Coils

In some embodiments, one or more of the resonant objects areself-resonant conducting coils. Referring to FIG. 2, a conducting wireof length l and cross-sectional radius a is wound into a helical coil ofradius r and height h (namely with N=√{square root over (l²−h²)}/2πrnumber of turns), surrounded by air. As described below, the wire hasdistributed inductance and distributed capacitance, and therefore itsupports a resonant mode of angular frequency ω. The nature of theresonance lies in the periodic exchange of energy from the electricfield within the capacitance of the coil, due to the charge distributionρ(x) across it, to the magnetic field in free space, due to the currentdistribution j(x) in the wire. In particular, the charge conservationequation ∇·j=iωρ implies that: (i) this periodic exchange is accompaniedby a π/2 phase-shift between the current and the charge densityprofiles, namely the energy U contained in the coil is at certain pointsin time completely due to the current and at other points in timecompletely due to the charge, and (ii) if ρ_(l)(x) and I(x) arerespectively the linear charge and current densities in the wire, wherex runs along the wire, q_(o)=½∫dx|ρ_(l)(x)| is the maximum amount ofpositive charge accumulated in one side of the coil (where an equalamount of negative charge always also accumulates in the other side tomake the system neutral) and I_(o)=max{|I(x)|} is the maximum positivevalue of the linear current distribution, then I_(o)=ωq₀. Then, one candefine an effective total inductance L and an effective totalcapacitance C of the coil through the amount of energy U inside itsresonant mode:

$\begin{matrix}{{\left. {U \equiv {\frac{1}{2}I_{o}^{2}L}}\Rightarrow L \right. = {\frac{\mu_{o}}{4\; \pi \; I_{o}^{2}}{\int{\int{{x}{x^{\prime}}\frac{{j(x)} \cdot {j\left( x^{\prime} \right)}}{{x - x^{\prime}}}}}}}},} & (2) \\{{\left. {U \equiv {\frac{1}{2}q_{o}^{2}\frac{1}{C}}}\Rightarrow\frac{1}{C} \right. = {\frac{1}{4\; {\pi ɛ}_{o}q_{o}^{2}}{\int{\int{{x}{x^{\prime}}\frac{{\rho (x)} \cdot {\rho \left( x^{\prime} \right)}}{{x - x^{\prime}}}}}}}},} & (3)\end{matrix}$

where μ_(o) and ∈_(o) are the magnetic permeability and electricpermittivity of free space. With these definitions, the resonant angularfrequency and the effective impedance are given by the common formulasω=1/√{square root over (LC)} and Z=√{square root over (L/C)}respectively.

Losses in this resonant system consist of ohmic (material absorption)loss inside the wire and radiative loss into free space. One can againdefine a total absorption resistance R_(abs) from the amount of powerabsorbed inside the wire and a total radiation resistance R_(rad) fromthe amount of power radiated due to electric- and magnetic-dipoleradiation:

$\begin{matrix}\left. {P_{abs} \equiv {\frac{1}{2}I_{o}^{2}R_{abs}}}\Rightarrow{R_{abs} \approx {\zeta_{c}{\frac{l}{2\; \pi \; a} \cdot \frac{I_{rms}^{2}}{I_{o}^{2}}}}} \right. & (4) \\{\left. {P_{rad} \equiv {\frac{1}{2}I_{o}^{2}R_{rad}}}\Rightarrow{R_{rad} \approx {\frac{\zeta_{o}}{6\; \pi}\left\lbrack {\left( \frac{\omega {p}^{2}}{c} \right) + \left( \frac{\omega \sqrt{m}}{c} \right)^{4}} \right\rbrack}} \right.,} & (5)\end{matrix}$

where c=1/√{square root over (μ₀/∈₀)} and ζ_(o)=/√{square root over(μ_(o)/∈_(o))} are the light velocity and light impedance in free space,the impedance ζ_(c) is ζ_(c)=1/σδ=√{square root over (μ₀ω/2σ)} with σthe conductivity of the conductor and δ the skin depth at the frequencyω, I_(rms) ²=1/l∫dx|I(x)|², p=∫dx rρ_(l)(x) is the electric-dipolemoment of the coil and m=½∫dx r×j(x) is the magnetic-dipole moment ofthe coil. For the radiation resistance formula Eq. (5), the assumptionof operation in the quasi-static regime (h,r<<λ=2πc/ω) has been used,which is the desired regime of a subwavelength resonance. With thesedefinitions, the absorption and radiation quality factors of theresonance are given by Q^(abs)=Z/R_(abs) and Q^(rad)=Z/R_(rad)respectively.

From Eq. (2)-(5) it follows that to determine the resonance parametersone simply needs to know the current distribution j in the resonantcoil. Solving Maxwell's equations to rigorously find the currentdistribution of the resonant electromagnetic eigenmode of aconducting-wire coil is more involved than, for example, of a standardLC circuit, and we can find no exact solutions in the literature forcoils of finite length, making an exact solution difficult. One could inprinciple write down an elaborate transmission-line-like model, andsolve it by brute force. We instead present a model that is (asdescribed below) in good agreement (˜5%) with experiment. Observing thatthe finite extent of the conductor forming each coil imposes theboundary condition that the current has to be zero at the ends of thecoil, since no current can leave the wire, we assume that the resonantmode of each coil is well approximated by a sinusoidal current profilealong the length of the conducting wire. We shall be interested in thelowest mode, so if we denote by x the coordinate along the conductor,such that it runs from −l/2 to +l/2, then the current amplitude profilewould have the form I(x)=I_(o) cos(πx/l), where we have assumed that thecurrent does not vary significantly along the wire circumference for aparticular x, a valid assumption provided a<<r. It immediately followsfrom the continuity equation for charge that the linear charge densityprofile should be of the form ρ_(l)(x)=ρ_(o) sin(πx/l), and thusq_(o)=∫₀ ^(l/2)dxρ_(o)|sin(πx/l)|=ρ_(o)l/π. Using these sinusoidalprofiles we find the so-called “self-inductance” L_(s) and“self-capacitance” C_(s) of the coil by computing numerically theintegrals Eq. (2) and (3); the associated frequency and effectiveimpedance are ω_(s) and Z_(s) respectively. The “self-resistances” R_(s)are given analytically by Eq. (4) and (5) using

${I_{rms}^{2} = {{\frac{1}{l}{\int_{{- l}/2}^{l/2}\ {{x}{{I_{o}{\cos \left( {\pi \; {x/l}} \right)}}}^{2}}}} = {\frac{1}{2}I_{o}^{2}}}},{{p} = {q_{o}\sqrt{\left( {\frac{2}{\pi}h} \right)^{2} + \left( {\frac{4\; N\; {\cos \left( {\pi \; N} \right)}}{\left( {{4\; N^{2}} - 1} \right)\pi}r} \right)^{2}}}}$

and

${{m} = {I_{o}\sqrt{\left( {\frac{2}{\pi}N\; \pi \; r^{2}} \right)^{2} + \left( {\frac{\begin{matrix}{{\cos \left( {\pi \; N} \right)\left( {{12\; N^{2}} - 1} \right)} -} \\{\sin \left( {\pi \; N} \right)\pi \; {N\left( {{4\; N^{2}} - 1} \right)}}\end{matrix}}{\left( {{16\; N^{4}} - {8\; N^{2}} + 1} \right)\pi}{hr}} \right)^{2}}}},$

and therefore the associated Q_(s) factors may be calculated.

The results for two particular embodiments of resonant coils withsubwavelength modes of λ_(s)/r≧70 (i.e. those highly suitable fornear-field coupling and well within the quasi-static limit) arepresented in Table 1. Numerical results are shown for the wavelength andabsorption, radiation and total loss rates, for the two different casesof subwavelength-coil resonant modes. Note that, for conductingmaterial, copper (σ=5.998·10̂−7 S/m) was used. It can be seen thatexpected quality factors at microwave frequencies are Q_(s) ^(abs)≧1000and Q_(s) ^(rad)≧5000.

TABLE 1 single coil λ_(s)/r f(MHz) Q_(s) ^(rad) Q_(s) ^(abs) Q_(s) =ω_(s)/2Γ_(s) r = 30 cm, h = 20 cm, a = 1 cm, N = 4 74.7 13.39 4164 81702758 r = 10 cm, h = 3 cm, a = 2 mm, N = 6 140 21.38 43919 3968 3639

Referring to FIG. 3, in some embodiments, energy is transferred betweentwo self-resonant conducting-wire coils. The electric and magneticfields are used to couple the different resonant conducting-wire coilsat a distance D between their centers. Usually, the electric couplinghighly dominates over the magnetic coupling in the system underconsideration for coils with h>>2r and, oppositely, the magneticcoupling highly dominates over the electric coupling for coils withh<<2r. Defining the charge and current distributions of two coils 1, 2respectively as ρ_(1,2) (x) and j_(1,2) (x), total charges and peakcurrents respectively as q_(1,2) and I_(1,2), and capacitances andinductances respectively as C_(1,2) and L_(1,2), which are the analogsof ρ(x), j(x), q_(o), I_(o), C and L for the single-coil case and aretherefore well defined, we can define their mutual capacitance andinductance through the total energy:

$\begin{matrix}{\begin{matrix}{U \equiv {U_{1} + U_{2} + {\frac{1}{2}{\left( {{q_{1}^{*}q_{2}} + {q_{2}^{*}q_{1}}} \right)/M_{C}}} +}} \\{\left. {\frac{1}{2}\left( {{I_{1}^{*}I_{2}} + {I_{2}^{*}I_{1}}} \right)M_{L}}\Rightarrow{1/M_{C}} \right.} \\{{= {\frac{1}{4\; \pi \; ɛ_{o}q_{1}q_{2}}{\int{\int{{x}{x^{\prime}}\frac{{\rho_{1}(x)} \cdot {\rho_{2}\left( x^{\prime} \right)}}{{x - x^{\prime}}}u}}}}},}\end{matrix}{{M_{L} = {\frac{\mu_{o}}{4\; \pi \; I_{1}I_{2}}{\int{\int{{x}{x^{\prime}}\frac{{j_{1}(x)} \cdot {j_{2}\left( x^{\prime} \right)}}{{x - x^{\prime}}}u}}}}},}} & (6)\end{matrix}$

where U₁=½q₁ ²/C₁=½I₁ ²L₁, U₂=½q₂ ²/C₂=½I₂ ²L₂ and the retardationfactor of u=exp(iω|x−x′|/c) inside the integral can been ignored in thequasi-static regime D<<λ of interest, where each coil is within the nearfield of the other. With this definition, the coupling coefficient isgiven by

$\begin{matrix}{\kappa = \left. {{\omega {\sqrt{C_{1}C_{2}}/2}\; M_{C}} + {\omega \; {M_{L}/2}\sqrt{L_{1}L_{2}}}}\Leftrightarrow Q_{\kappa}^{- 1} \right.} \\{= {\left( {M_{C}/\sqrt{C_{1}C_{2}}} \right)^{- 1} + {\left( {\sqrt{L_{1}L_{2}}/M_{L}} \right)^{- 1}.}}}\end{matrix}$

Therefore, to calculate the coupling rate between two self-resonantcoils, again the current profiles are needed and, by using again theassumed sinusoidal current profiles, we compute numerically from Eq. (6)the mutual capacitance M_(C,s) and inductance M_(L,s) between twoself-resonant coils at a distance D between their centers, and thusQ_(κ,s) is also determined.

TABLE 2 pair of coils D/r Q = ω/2Γ Q_(κ) = ω/2κ κ/Γ r = 30 cm, h = 20cm, 3 2758 38.9 70.9 a = 1 cm, N = 4 5 2758 139.4 19.8 λ/r ≈ 75 7 2758333.0 8.3 Q_(s) ^(abs) ≈ 8170, Q_(s) ^(rad) ≈ 4164 10 2758 818.9 3.4 r =10 cm, h = 3 cm, 3 3639 61.4 59.3 a = 2 mm, N = 6 5 3639 232.5 15.7 λ/r≈ 140 7 3639 587.5 6.2 Q_(s) ^(abs) ≈ 3968, Q_(s) ^(rad) ≈ 43919 10 36391580 2.3

Referring to Table 2, relevant parameters are shown for exemplaryembodiments featuring pairs or identical self resonant coils. Numericalresults are presented for the average wavelength and loss rates of thetwo normal modes (individual values not shown), and also the couplingrate and figure-of-merit as a function of the coupling distance D, forthe two cases of modes presented in Table 1. It can be seen that formedium distances D/r=10−3 the expected coupling-to-loss ratios are inthe range κ/Γ˜2−70.

Capacitively-Loaded Conducting Loops or Coils

In some embodiments, one or more of the resonant objects arecapacitively-loaded conducting loops or coils. Referring to FIG. 4 ahelical coil with N turns of conducting wire, as described above, isconnected to a pair of conducting parallel plates of area A spaced bydistance d via a dielectric material of relative permittivity ∈, andeverything is surrounded by air (as shown, N=1 and h=0). The plates havea capacitance C_(p)=∈₀∈A/d, which is added to the distributedcapacitance of the coil and thus modifies its resonance. Note however,that the presence of the loading capacitor modifies significantly thecurrent distribution inside the wire and therefore the total effectiveinductance L and total effective capacitance C of the coil are differentrespectively from L_(s) and C_(s), which are calculated for aself-resonant coil of the same geometry using a sinusoidal currentprofile. Since some charge is accumulated at the plates of the externalloading capacitor, the charge distribution ρ inside the wire is reduced,so C<C_(s), and thus, from the charge

conservation equation, the current distribution j flattens out, soL>L_(s). The resonant frequency for this system is ω=1/√{square rootover (L/(C+C_(p)))}<ω_(s)=1/√{square root over (L_(s)C_(s))}, andI(x)→I_(o) cos(πx/l)

C→C_(s)

ω→ω_(s), as C_(p)→0.

In general, the desired CMT parameters can be found for this system, butagain a very complicated solution of Maxwell's Equations is required.Instead, we will analyze only a special case, where a reasonable guessfor the current distribution can be made. When C_(p)>>C_(s)>C, thenω≈1/√{square root over (LC_(p))}<<ω_(s) and Z≈√{square root over(L/C_(p))}<<Z_(s), while all the charge is on the plates of the loadingcapacitor and thus the current distribution is constant along the wire.This allows us now to compute numerically L from Eq. (2). In the caseh=0 and N integer, the integral in Eq. (2) can actually be computedanalytically, giving the formula L=μ^(o)r[ln(8r/a)−2]N². Explicitanalytical formulas are again available for R from Eq. (4) and (5),since I_(rms)=I_(o), |p|≈0 and |m|=I_(o)Nπr² (namely only themagnetic-dipole term is contributing to radiation), so we can determinealso Q^(abs)=ωL/R_(abs) and Q^(rad)=ωL/R_(rad). At the end of thecalculations, the validity of the assumption of constant current profileis confirmed by checking that indeed the condition C_(p)>>C_(s)

ω<<ω_(s) is satisfied. To satisfy this condition, one could use a largeexternal capacitance, however, this would usually shift the operationalfrequency lower than the optimal frequency, which we will determineshortly; instead, in typical embodiments, one often prefers coils withvery small self-capacitance C_(s) to begin with, which usually holds,for the types of coils under consideration, when N=1, so that theself-capacitance comes from the charge distribution across the singleturn, which is almost always very small, or when N>1 and h>>2Na, so thatthe dominant self-capacitance comes from the charge distribution acrossadjacent turns, which is small if the separation between adjacent turnsis large.

The external loading capacitance C_(p) provides the freedom to tune theresonant frequency (for example by tuning A or d). Then, for theparticular simple case h=0, for which we have analytical formulas, thetotal Q=ωL/(R_(abs)+R_(rad)) becomes highest at the optimal frequency

$\begin{matrix}{{\overset{\sim}{\omega} = \left\lbrack {\frac{c^{4}}{\pi}{\sqrt{\frac{ɛ_{o}}{2\; \sigma}} \cdot \frac{1}{a\; N\; r^{3}}}} \right\rbrack^{\frac{2}{7}}},} & (7)\end{matrix}$

reaching the value

$\begin{matrix}{\overset{\sim}{Q} = {\frac{6}{7\; \pi}{\left( {2\; \pi^{2}\eta_{o}\frac{\sigma \; a^{2}N^{2}}{r}} \right)^{\frac{3}{7}} \cdot {\left\lbrack {{\ln\left( \frac{8\; r}{a} \right)} - 2} \right\rbrack.}}}} & (8)\end{matrix}$

At lower frequencies it is dominated by ohmic loss and at higherfrequencies by radiation. Note, however, that the formulas above areaccurate as long as {tilde over (ω)}<<ω_(s) and, as explained above,this holds almost always when N=1, and is usually less accurate whenN>1, since h=0 usually implies a large self-capacitance. A coil withlarge h can be used, if the self-capacitance needs to be reducedcompared to the external capacitance, but then the formulas for L and{tilde over (ω)}, {tilde over (Q)} are again less accurate. Similarqualitative behavior is expected, but a more complicated theoreticalmodel is needed for making quantitative predictions in that case.

The results of the above analysis for two embodiments of subwavelengthmodes of λ/r≧70 (namely highly suitable for near-field coupling and wellwithin the quasi-static limit) of coils with N=1 and h=0 at the optimalfrequency Eq. (7) are presented in Table 3. To confirm the validity ofconstant-current assumption and the resulting analytical formulas,mode-solving calculations were also performed using another completelyindependent method: computational 3D finite-element frequency-domain(FEFD) simulations (which solve Maxwell's Equations in frequency domainexactly apart for spatial discretization) were conducted, in which theboundaries of the conductor were modeled using a complex impedanceζ_(e)=√{square root over (μ_(o)ω/2σ)} boundary condition, valid as longas ζ_(e)/ζ_(o)<<1 (<10⁻⁵ for copper in the microwave). Table 3 showsNumerical FEFD (and in parentheses analytical) results for thewavelength and absorption, radiation and total loss rates, for twodifferent cases of subwavelength-loop resonant modes. Note that forconducting material copper (σ=5.998·10⁷S/m) was used. (The specificparameters of the plot in FIG. 4 are highlighted with bold in thetable.) The two methods (analytical and computational) are in very goodagreement and show that, in some embodiments, the optimal frequency isin the low-MHz microwave range and the expected quality factors areQ_(abs)≧1000 and Q^(rad)≧10000.

TABLE 3 single coil λ/r f(MHz) Q^(rad) Q^(abs) Q = ω/2Γ r = 30 cm, a = 2cm 111.4 (112.4) 8.976 (8.897) 29546 (30512) 4886 (5117) 4193 (4381) ε =10, A = 138 cm 2 , d = 4 mm r = 10 cm, a = 2 mm 69.7 (70.4) 43.04(42.61) 10702 (10727) 1545 (1604) 1350 (1395) ε = 10, A = 3.14 cm², d =1 mm

Referring to FIG. 5, in some embodiments, energy is transferred betweentwo capacitively-loaded coils. For the rate of energy transfer betweentwo capacitively-loaded coils 1 and 2 at distance D between theircenters, the mutual inductance M_(L) can be evaluated numerically fromEq. (6) by using constant current distributions in the case ω<<ω_(s). Inthe case h=0, the coupling is only magnetic and again we have ananalytical formula, which, in the quasi-static limit r<<D<<λ and for therelative orientation shown in FIG. 4, is M_(L)≈πμ_(o)/2·(r₁r₂)²N₁N₂/D³,which means that Q_(κ)∝(D/√{square root over (r₁r₂)})³ is independent ofthe frequency ω and the number of turns N₁, N₂ Consequently, theresultant coupling figure-of-merit of interest is

$\begin{matrix}{{\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} = {\frac{\sqrt{Q_{1}Q_{2}}}{Q_{\kappa}} \approx {\left( \frac{\sqrt{r_{1}r_{2}}}{D} \right)^{3} \cdot \frac{\pi^{2}\eta_{o}{\frac{\sqrt{r_{1}r_{2}}}{\lambda} \cdot N_{1}}N_{2}}{\prod\limits_{{j = 1},2}\; \begin{pmatrix}{{{\sqrt{\frac{\pi \; \eta_{o}}{\lambda\sigma}} \cdot \frac{r_{j}}{a_{j}}}N_{j}} +} \\{\frac{8}{3}\pi^{5}{\eta_{o}\left( \frac{r_{j}}{\lambda} \right)}^{4}N_{j}^{2}}\end{pmatrix}^{\frac{1}{2}}}}}},} & (9)\end{matrix}$

which again is more accurate for N₁=N₂=1.

From Eq. (9) it can be seen that the optimal frequency {tilde over (ω)},where the figure-of-merit is maximized to the value (√{square root over(Q₁Q₂)}/Q_(κ)), is that where √{square root over (Q₁Q₂)} is maximized,since Q_(κ) does not depend on frequency (at least for the distancesD<<λ of interest for which the quasi-static approximation is stillvalid). Therefore, the optimal frequency is independent of the distanceD between the two coils and lies between the two frequencies where thesingle-coil Q₁ and Q₂ peak. For same coils, it is given by Eq. (7) andthen the figure-of-merit Eq. (9) becomes

$\begin{matrix}{\overset{\sim}{\left( \frac{k}{\Gamma} \right)} = {\frac{\overset{\sim}{Q}}{Q_{\kappa}} \approx {{\left( \frac{r}{D} \right)^{3} \cdot \frac{3}{7}}{\left( {2\; \pi^{2}\eta_{o}\frac{\sigma \; a^{2}N^{2}}{r}} \right)^{\frac{3}{7}}.}}}} & (10)\end{matrix}$

Typically, one should tune the capacitively-loaded conducting loops orcoils, so that their angular eigenfrequencies are close to {tilde over(ω)} within {tilde over (Γ)}, which is half the angular frequency widthfor which √{square root over (Q₁Q₂)}/Q_(κ)>(√{square root over(Q₁Q₂)}/Q_(κ))/2.

Referring to Table 4, numerical FEFD and, in parentheses, analyticalresults based on the above are shown for two systems each composed of amatched pair of the loaded coils described in Table 3. The averagewavelength and loss rates are shown along with the coupling rate andcoupling to loss ratio figure-of-merit κ/Γ as a function of the couplingdistance D, for the two cases. Note that the average numerical Γ^(rad)shown are again slightly different from the single-loop value of FIG. 3,analytical results for Γ^(rad) are not shown but the single-loop valueis used. (The specific parameters corresponding to the plot in FIG. 5are highlighted with bold in the table.) Again we chose N=1 to make theconstant-current assumption a good one and computed M_(L) numericallyfrom Eq. (6). Indeed the accuracy can be confirmed by their agreementwith the computational FEFD mode-solver simulations, which give κthrough the frequency splitting (=2κ) of the two normal modes of thecombined system. The results show that for medium distances D/r=10−3 theexpected coupling-to-loss ratios are in the range κ/Γ˜0.5-50.

TABLE 4 pair of coils D/r Q^(rad) Q = ω/2Γ Q_(κ)= ω/2κ κ/Γ r = 30 cm, a= 2 cm 3 30729 4216 62.6 (63.7) 67.4 (68.7) ε = 10, A = 138 cm 2 , d = 4mm 5 29577 4194 235 (248) 17.8 (17.6) λ/r ≈ 112 7 29128 4185 589 (646)7.1 (6.8) Q^(abs) ≈ 4886 10 28833 4177 1539 (1828) 2.7 (2.4) r = 10 cm,a = 2 mm 3 10955 1355 85.4 (91.3) 15.9 (15.3) ε = 10, A = 3.14 cm², d =1 mm 5 10740 1351 313 (356) 4.32 (3.92) λ/r ≈ 70 7 10759 1351 754 (925)1.79 (1.51) Q^(abs) ≈ 1546 10 10756 1351 1895 (2617) 0.71 (0.53)

Optimization of √{square root over (Q₁Q₂)}/Q_(κ)

In some embodiments, the results above can be used to increase oroptimize the performance of a wireless energy transfer system whichemploys capacitively-loaded coils. For example, the scaling of Eq. (10)with the different system parameters one sees that to maximize thesystem figure-of-merit κ/Γ one can, for example:

-   -   Decrease the resistivity of the conducting material. This can be        achieved, for example, by using good conductors (such as copper        or silver) and/or lowering the temperature. At very low        temperatures one could use also superconducting materials to        achieve extremely good performance.    -   Increase the wire radius a. In typical embodiments, this action        is limited by physical size considerations. The purpose of this        action is mainly to reduce the resistive losses in the wire by        increasing the cross-sectional area through which the electric        current is flowing, so one could alternatively use also a Litz        wire or a ribbon instead of a circular wire.    -   For fixed desired distance D of energy transfer, increase the        radius of the loop r. In typical embodiments, this action is        limited by physical size considerations.    -   For fixed desired distance vs. loop-size ratio D/r, decrease the        radius of the loop r. In typical embodiments, this action is        limited by physical size considerations.    -   Increase the number of turns N. (Even though Eq. (10) is        expected to be less accurate for N>1, qualitatively it still        provides a good indication that we expect an improvement in the        coupling-to-loss ratio with increased N.) In typical        embodiments, this action is limited by physical size and        possible voltage considerations, as will be discussed in        following sections.    -   Adjust the alignment and orientation between the two coils. The        figure-of-merit is optimized when both cylindrical coils have        exactly the same axis of cylindrical symmetry (namely they are        “facing” each other). In some embodiments, particular mutual        coil angles and orientations that lead to zero mutual inductance        (such as the orientation where the axes of the two coils are        perpendicular) should be avoided.    -   Finally, note that the height of the coil h is another available        design parameter, which has an impact to the performance similar        to that of its radius r, and thus the design rules are similar.

The above analysis technique can be used to design systems with desiredparameters. For example, as listed below, the above described techniquescan be used to determine the cross sectional radius a of the wire whichone should use when designing as system two same single-turn loops witha given radius in order to achieve a specific performance in terms ofκ/Γ at a given D/r between them, when the material is copper(σ=5.998·10⁷ S/m):

D/r=5, κ/Γ≧10, r=30 cm

a≧9 mm

D/r=5, κ/Γ≧10, r=5 cm

a≧3.7 mm

D/r=5, κ/Γ≧20, r=30 cm

a≧20 mm

D/r=5, κ/Γ≧20, r=5 cm

a≧8.3 mm

D/r=10, κ/Γ≧1, r=30 cm

a≧7 mm

D/r=10, κ/Γ≧1, r=5 cm

a≧2.8 mm

D/r=10, κ/Γ≧3, r=30 cm

a>25 mm

D/r=10, κ/Γ≧3, r=5 cm

a≧10 mm

Similar analysis can be done for the case of two dissimilar loops. Forexample, in some embodiments, the device under consideration is veryspecific (e.g. a laptop or a cell phone), so the dimensions of thedevice object (r_(d), h_(d), a_(d), N_(d)) are very restricted. However,in some such embodiments, the restrictions on the source object (r_(s),h_(s), a_(s), N_(s)) are much less, since the source can, for example,be placed under the floor or on the ceiling. In such cases, the desireddistance is often well defined, based on the application (e.g. D˜1 m forcharging a laptop on a table wirelessly from the floor). Listed beloware examples (simplified to the case N_(s)=N_(d)=1 and h_(s)=h_(d)=0) ofhow one can vary the dimensions of the source object to achieve thedesired system performance in terms of κ/√{square root over(Γ_(s)Γ_(d))}, when the material is again copper (σ=5.998·10⁷S/m):

D=1.5 m, κ/√{square root over (Γ_(s)Γ_(d))}≧15, r_(d)=30 cm, a_(d)=6 mm

r_(s)=1.158 m, a_(s)≧5 mm

D=1.5 m, κ/√{square root over (Γ_(s)Γ_(d))}≧30, r_(d)=30 cm, a_(d)=6 mm

r_(s)=1.15 m, a_(s)≧33 mm

D=1.5 m, κ/√{square root over (Γ_(s)Γ_(d))}≧1, r_(d)=5 cm, a_(d)=4 mm

r_(s)=1.119 m, a_(s)≧7 mm

D=1.5 m, κ/√{square root over (Γ_(s)Γ_(d))}≧2, r_(d)=5 cm, a_(d)=4 mm

r_(s)=1.119 m, a_(s)≧52 mm

D=2 m, κ/√{square root over (Γ_(s)Γ_(d))}≧10, r_(d)=30 cm, a_(d)=6 mm

r_(s)=1.518 m, a_(s)≧7 mm

D=2 m, κ/√{square root over (Γ_(s)Γ_(d))}≧20, r_(d)=30 cm, a_(d)=6 mm

r_(s)=1.514 m, a_(s)≧50 mm

D=2 m, κ/√{square root over (Γ_(s)Γ_(d))}≧0.5, r_(d)=5 cm, a_(d)=4 mm

r_(s)=1.491 m, a_(s)≧5 mm

D=2 m, κ/√{square root over (Γ_(s)Γ_(d))}≧1, r_(d)=5 cm, a_(d)=4 mm

r_(s)=1.491 m, a_(s)>36 mm

Optimization of Q_(κ)

As will be described below, in some embodiments the quality factor Q ofthe resonant objects is limited from external perturbations and thusvarying the coil parameters cannot lead to improvement in Q. In suchcases, one may opt to increase the coupling to loss ratiofigure-of-merit by decreasing Q_(κ) (i.e. increasing the coupling). Thecoupling does not depend on the frequency and the number of turns.Therefore, the remaining degrees of freedom are:

-   -   Increase the wire radii a₁ and a₂. In typical embodiments, this        action is limited by physical size considerations.    -   For fixed desired distance D of energy transfer, increase the        radii of the coils r₁ and r₂. In typical embodiments, this        action is limited by physical size considerations.    -   For fixed desired distance vs. coil-sizes ratio D/√{square root        over (r₁r₂)}, only the weak (logarithmic) dependence of the        inductance remains, which suggests that one should decrease the        radii of the coils r₁ and r₂. In typical embodiments, this        action is limited by physical size considerations.    -   Adjust the alignment and orientation between the two coils. In        typical embodiments, the coupling is optimized when both        cylindrical coils have exactly the same axis of cylindrical        symmetry (namely they are “facing” each other). Particular        mutual coil angles and orientations that lead to zero mutual        inductance (such as the orientation where the axes of the two        coils are perpendicular) should obviously be avoided.    -   Finally, note that the heights of the coils h₁ and h₂ are other        available design parameters, which have an impact to the        coupling similar to that of their radii r₁ and r₂, and thus the        design rules are similar.

Further practical considerations apart from efficiency, e.g. physicalsize limitations, will be discussed in detail below.

It is also important to appreciate the difference between the abovedescribed resonant-coupling inductive scheme and the well-knownnon-resonant inductive scheme for energy transfer. Using CMT it is easyto show that, keeping the geometry and the energy stored at the sourcefixed, the resonant inductive mechanism allows for ˜Q² (˜10⁶) times morepower delivered for work at the device than the traditional non-resonantmechanism. This is why only close-range contact-less medium-power (˜W)transfer is possible with the latter, while with resonance eitherclose-range but large-power (˜kW) transfer is allowed or, as currentlyproposed, if one also ensures operation in the strongly-coupled regime,medium-range and medium-power transfer is possible. Capacitively-loadedconducting loops are currently used as resonant antennas (for example incell phones), but those operate in the far-field regime with D/r>>1,r/λ˜1, and the radiation Q's are intentionally designed to be small tomake the antenna efficient, so they are not appropriate for energytransfer.

Inductively-Loaded Conducting Rods

A straight conducting rod of length 2h and cross-sectional radius a hasdistributed capacitance and distributed inductance, and therefore itsupports a resonant mode of angular frequency ω. Using the sameprocedure as in the case of self-resonant coils, one can define aneffective total inductance L and an effective total capacitance C of therod through formulas (2) and (3). With these definitions, the resonantangular frequency and the effective impedance are given again by thecommon formulas ω=1/√{square root over (LC)} and Z=√{square root over(L/C)} respectively. To calculate the total inductance and capacitance,one can assume again a sinusoidal current profile along the length ofthe conducting wire. When interested in the lowest mode, if we denote byx the coordinate along the conductor, such that it runs from −h to +h,then the current amplitude profile would have the form I(x)=I_(o)cos(πx/2h), since it has to be zero at the open ends of the rod. This isthe well-known half-wavelength electric dipole resonant mode.

In some embodiments, one or more of the resonant objects areinductively-loaded conducting rods. A straight conducting rod of length2h and cross-sectional radius a, as in the previous paragraph, is cutinto two equal pieces of length h, which are connected via a coilwrapped around a magnetic material of relative permeability, andeverything is surrounded by air. The coil has an inductance L_(C), whichis added to the distributed inductance of the rod and thus modifies itsresonance. Note however, that the presence of the center-loadinginductor modifies significantly the current distribution inside the wireand therefore the total effective inductance L and total effectivecapacitance C of the rod are different respectively from L, and C_(s),which are calculated for a self-resonant rod of the same total lengthusing a sinusoidal current profile, as in the previous paragraph. Sincesome current is running inside the coil of the external loadinginductor, the current distribution j inside the rod is reduced, soL<L_(s), and thus, from the charge conservation equation, the linearcharge distribution ρ_(l) flattens out towards the center (beingpositive in one side of the rod and negative in the other side of therod, changing abruptly through the inductor), so C>C_(s). The resonantfrequency for this system is ω=1/√{square root over((L+L_(C))C)}<ω_(s)=1/√{square root over (L_(s)C_(s))}, and I(x)→I_(o)cos(πx/2h)

L→L_(s)

ω→ω_(s), as L_(c)→0.

In general, the desired CMT parameters can be found for this system, butagain a very complicated solution of Maxwell's Equations is required.Instead, we will analyze only a special case, where a reasonable guessfor the current distribution can be made. When L_(c)>>L_(s)>L, thenω≈1/√{square root over (L_(C)C)}<<ω_(s) and Z≈√{square root over(L_(c)/C)}>>Z_(s), while the current distribution is triangular alongthe rod (with maximum at the center-loading inductor and zero at theends) and thus the charge distribution is positive constant on one halfof the rod and equally negative constant on the other side of the rod.This allows us now to compute numerically C from Eq. (3). In this case,the integral in Eq. (3) can actually be computed analytically, givingthe formula 1/C=1/(π∈_(o)h)[ln(h/a)−1]. Explicit analytical formulas areagain available for R from Eq. (4) and (5), since I_(rms)=I_(o),|p|=q_(o)h and |m|=0 (namely only the electric-dipole term iscontributing to radiation), so we can determine also Q^(abs)=1/ωCR_(abs)and Q^(rad)=1/ωCR_(rad). At the end of the calculations, the validity ofthe assumption of triangular current profile is confirmed by checkingthat indeed the condition L_(c)>>L_(s)

ω<<ω_(s) is satisfied. This condition is relatively easily satisfied,since typically a conducting rod has very small self-inductance L_(s) tobegin with.

Another important loss factor in this case is the resistive loss insidethe coil of the external loading inductor L_(c) and it depends on theparticular design of the inductor. In some embodiments, the inductor ismade of a Brooks coil, which is the coil geometry which, for fixed wirelength, demonstrates the highest inductance and thus quality factor. TheBrooks coil geometry has N_(Bc) turns of conducting wire ofcross-sectional radius a_(Bc) wrapped around a cylindrically symmetriccoil former, which forms a coil with a square cross-section of sider_(Bc), where the inner side of the square is also at radius r_(Bc) (andthus the outer side of the square is at radius 2r_(Bc)), thereforeN_(Bc)≈(r_(Bc)/2a_(Bc))². The inductance of the coil is thenL_(c)=2.0285μ_(o)r_(Bc)N_(Bc) ²≈2.0285μ_(o)r_(Bc) ⁵/8a_(Bc) ⁴ and itsresistance

${R_{c} \approx {\frac{1}{\sigma}\frac{l_{Bc}}{\pi \; a_{Bc}^{2}}\sqrt{1 + {\frac{\mu_{o}\omega \; \sigma}{2}\left( \frac{a_{Bc}}{2} \right)^{2}}}}},$

where the total wire length is l_(Bc)≈2π(3r_(Bc)/2)N_(Bc)≈3πr_(Bc) ³/4a_(Bc) ³ and we have used an approximate square-root law for thetransition of the resistance from the dc to the ac limit as the skindepth varies with frequency.

The external loading inductance L_(c) provides the freedom to tune theresonant frequency. (For example, for a Brooks coil with a fixed sizer_(Bc), the resonant frequency can be reduced by increasing the numberof turns NBC by decreasing the wire cross-sectional radius a_(Bc). Thenthe desired resonant angular frequency ω=1/ √{square root over (L_(c)C)}is achieved for a_(Bc)≈(2.0285μ_(o)r_(Bc) ⁵ω²C)^(1/4) and the resultingcoil quality factor is

$\left. {Q_{c} \approx {0.169\mu_{o}\sigma \; r_{Bc}^{2}{\omega/\sqrt{1 + {\omega^{2}\mu_{o}\sigma \sqrt{2.0285{\mu_{o}\left( {r_{Bc}/4} \right)}^{5}C}}}}}} \right).$

Then, for the particular simple case L_(c)>>L_(s), for which we haveanalytical formulas, the total Q=1/ωC(R_(c)+R_(abs)+R_(rad)) becomeshighest at some optimal frequency a {tilde over (ω)}, reaching the value{tilde over (Q)}, both determined by the loading-inductor specificdesign. (For example, for the Brooks-coil procedure described above, atthe optimal frequency {tilde over (Q)}≈Q_(c)≈0.8(μ_(o)σ²r_(Bc)³/C)^(1/4)) At lower frequencies it is dominated by ohmic loss insidethe inductor coil and at higher frequencies by radiation. Note, again,that the above formulas are accurate as long as {tilde over (ω)}<<ω_(s)and, as explained above, this is easy to design for by using a largeinductance.

The results of the above analysis for two embodiments, using Brookscoils, of subwavelength modes of λ/h≧200 (namely highly suitable fornear-field coupling and well within the quasi-static limit) at theoptimal frequency {tilde over (ω)} are presented in Table 5. Table 5shows in parentheses (for similarity to previous tables) analyticalresults for the wavelength and absorption, radiation and total lossrates, for two different cases of subwavelength-loop resonant modes.Note that for conducting material copper (σ=5.998·10⁷S/m) was used. Theresults show that, in some embodiments, the optimal frequency is in thelow-MHz microwave range and the expected quality factors areQ^(abs)≧1000 and Q^(rad)>100000.

TABLE 5 single rod λ/h f(MHz) Q^(rad) Q^(abs) Q = ω/2Γ h = 30 cm, a = 2cm (403.8) (2.477) (2.72 * 10⁶) (7400) (7380) μ = 1, r_(Bc) = 2 cm,a_(Bc) = 0.88 mm, N_(Bc) = 129 h = 10 cm, a = 2 mm (214.2) (14.010)(6.92 * 10⁵) (3908) (3886) μ = 1, r_(Bc) = 5 mm, a_(Bc) = 0.25 mm,

In some embodiments, energy is transferred between twoinductively-loaded rods. For the rate of energy transfer between twoinductively-loaded rods 1 and 2 at distance D between their centers, themutual capacitance M_(C) can be evaluated numerically from Eq. (6) byusing triangular current distributions in the case ω<<ω_(s). In thiscase, the coupling is only electric and again we have an analyticalformula, which, in the quasi-static limit h<<D<<λ and for the relativeorientation such that the two rods are aligned on the same axis, is1/M_(C)≈1/2π∈_(o)·(h₁h₂)²/D³, which means that Q_(κ)∝(D/√{square rootover (h₁h₂)})³ is independent of the frequency ω. Consequently, one canget the resultant coupling figure-of-merit of interest

$\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} = {\frac{\sqrt{Q_{1}Q_{2}}}{Q_{\kappa}}.}$

It can be seen that the optimal frequency {tilde over (ω)}A, where thefigure-of-merit is maximized to the value (√{square root over(Q₁Q₂)}/Q_(κ)), is that where √{square root over (Q₁Q₂)} is maximized,since Q_(κ) does not depend on frequency (at least for the distancesD<<λ of interest for which the quasi-static approximation is stillvalid). Therefore, the optimal frequency is independent of the distanceD between the two rods and lies between the two frequencies where thesingle-rod Q₁ and Q₂ peak. Typically, one should tune theinductively-loaded conducting rods, so that their angulareigenfrequencies are close to {tilde over (ω)} within {tilde over (Γ)},which is half the angular frequency width for which √{square root over(Q₁Q₂)}/Q_(κ)> √{square root over (Q₁Q₂)}/Q_(κ))/2.

Referring to Table 6, in parentheses (for similarity to previous tables)analytical results based on the above are shown for two systems eachcomposed of a matched pair of the loaded rods described in Table 5. Theaverage wavelength and loss rates are shown along with the coupling rateand coupling to loss ratio figure-of-merit κ/Γ as a function of thecoupling distance D, for the two cases. Note that for Γ^(rad) thesingle-rod value is used. Again we chose L_(c)>>L_(s) to make thetriangular-current assumption a good one and computed M_(C) numericallyfrom Eq. (6). The results show that for medium distances D/h=10−3 theexpected coupling-to-loss ratios are in the range κ/Γ˜0.5-100.

TABLE 6 pair of rods D/h Q_(κ) = ω/2κ κ/Γ h = 30 cm, a = 2 cm 3   (70.3) (105.0) μ = 1, r_(Bc) = 2 cm, 5  (389) (19.0) a_(Bc) = 0.88mm, N_(Bc) = 129 λ/h ≈ 404 7 (1115) (6.62) Q ≈ 7380 10 (3321) (2.22) h =10 cm, a = 2 mm 3  (120) (32.4) μ = 1, r_(Bc) = 5 mm, 5  (664) (5.85)a_(Bc) = 0.25 mm, N_(Bc) = 103 λ/h ≈ 214 7 (1900) (2.05) Q ≈ 3886 10(5656) (0.69)

Dielectric Disks

In some embodiments, one or more of the resonant objects are dielectricobjects, such as disks. Consider a two dimensional dielectric diskobject, as shown in FIG. 6, of radius r and relative permittivity Csurrounded by air that supports high-Q “whispering-gallery” resonantmodes. The loss mechanisms for the energy stored inside such a resonantsystem are radiation into free space and absorption inside the diskmaterial. High-Q_(rad) and long-tailed subwavelength resonances can beachieved when the dielectric permittivity C is large and the azimuthalfield variations are slow (namely of small principal number m). Materialabsorption is related to the material loss tangent: Q_(abs)˜Re{∈}/Im{∈}.Mode-solving calculations for this type of disk resonances wereperformed using two independent methods: numerically, 2Dfinite-difference frequency-domain (FDFD) simulations (which solveMaxwell's Equations in frequency domain exactly apart for spatialdiscretization) were conducted with a resolution of 30 pts/r;analytically, standard separation of variables (SV) in polar coordinateswas used.

TABLE 7 single disk λ/r Q^(abs) Q^(rad) Q Re{ε} = 147.7, m = 2 20.01(20.00) 10103 (10075) 1988 (1992) 1661 (1663) Re{ε} = 65.6, m = 3 9.952(9.950) 10098 (10087) 9078 (9168) 4780 (4802)The results for two TE-polarized dielectric-disk subwavelength modes ofλ/r≧10 are presented in Table 7. Table 7 shows numerical FDFD (and inparentheses analytical SV) results for the wavelength and absorption,radiation and total loss rates, for two different cases ofsubwavelength-disk resonant modes. Note that disk-material loss-tangentIm{∈}/Re{∈}=10⁻⁴ was used. (The specific parameters corresponding to theplot in FIG. 6. are highlighted with bold in the table.) The two methodshave excellent agreement and imply that for a properly designed resonantlow-loss-dielectric object values of Q_(rad)≧2000 and Q_(abs)˜10000 areachievable. Note that for the 3D case the computational complexity wouldbe immensely increased, while the physics would not be significantlydifferent. For example, a spherical object of ∈=147.7 has a whisperinggallery mode with m=2, Qrad=13962, and λ/r=17.

The required values of ∈, shown in Table 7, might at first seemunrealistically large. However, not only are there in the microwaveregime (appropriate for approximately meter-range coupling applications)many materials that have both reasonably high enough dielectricconstants and low losses (e.g. Titania, Barium tetratitanate, Lithiumtantalite etc.), but also ∈ could signify instead the effective index ofother known subwavelength surface-wave systems, such as surface modes onsurfaces of metallic materials or plasmonic (metal-like, negative-∈)materials or metallo-dielectric photonic crystals or plasmono-dielectricphotonic crystals.

To calculate now the achievable rate of energy transfer between twodisks 1 and 2, as shown in FIG. 7 we place them at distance D betweentheir centers. Numerically, the FDFD mode-solver simulations give κthrough the frequency splitting (=2κ) of the normal modes of thecombined system, which are even and odd superpositions of the initialsingle-disk modes; analytically, using the expressions for theseparation-of-variables eigenfields E_(1,2(r)) CMT gives κ throughκ=ω₁/2·∫d³r∈₂(r)E₂*(r)E₁(r)/∫d³r∈(r)|E₁(r)|² where ∈_(j)(r) and ∈(r) arethe dielectric functions that describe only the disk j (minus theconstant ∈_(o) background) and the whole space respectively. Then, formedium distances D/r=10−3 and for non-radiative coupling such thatD<2r_(c), where r_(c)=mλ/2π is the radius of the radiation caustic, thetwo methods agree very well, and we finally find, as shown in Table 8,coupling-to-loss ratios in the range κ/Γ˜1-50. Thus, for the analyzedembodiments, the achieved figure-of-merit values are large enough to beuseful for typical applications, as discussed below.

TABLE 8 two disks D/r Q^(rad) Q = ω/2Γ ω/2κ κ/Γ Re{ε} = 147.7, 3 24781989 46.9 (47.5) 42.4 (35.0) m = 2 λ/r ≈ 20 5 2411 1946 298.0 (298.0)6.5 (5.6) Q^(abs) ≈ 10093 7 2196 1804 769.7 (770.2) 2.3 (2.2) 10 20171681 1714 (1601) 0.98 (1.04) Re{ε} = 65.6, 3 7972 4455 144 (140) 30.9(34.3) m = 3 λ/r ≈ 10 5 9240 4824 2242 (2083) 2.2 (2.3) Q^(abs) ≈ 100967 9187 4810 7485 (7417) 0.64 (0.65)

Note that even though particular embodiments are presented and analyzedabove as examples of systems that use resonant electromagnetic couplingfor wireless energy transfer, those of self-resonant conducting coils,capacitively-loaded resonant conducting coils and resonant dielectricdisks, any system that supports an electromagnetic mode with itselectromagnetic energy extending much further than its size can be usedfor transferring energy. For example, there can be many abstractgeometries with distributed capacitances and inductances that supportthe desired kind of resonances. In any one of these geometries, one canchoose certain parameters to increase and/or optimize √{square root over(Q₁Q₂)}/Q_(κ) or, if the Q's are limited by external factors, toincrease and/or optimize for Q_(κ).

System Sensitivity to Extraneous Objects

In general, the overall performance of particular embodiment of theresonance-based wireless energy-transfer scheme depends strongly on therobustness of the resonant objects' resonances. Therefore, it isdesirable to analyze the resonant objects' sensitivity to the nearpresence of random non-resonant extraneous objects. One appropriateanalytical model is that of “perturbation theory” (PT), which suggeststhat in the presence of an extraneous object e the field amplitude a₁(t)inside the resonant object 1 satisfies, to first order:

$\begin{matrix}{\frac{a_{1}}{t} = {{{- {\left( {\omega_{1} - {\; \Gamma_{1}}} \right)}}a_{1}} + {{\left( {\kappa_{11 - e} + {\; \Gamma_{1 - e}}} \right)}a_{1}}}} & (11)\end{matrix}$

where again ω₁ is the frequency and Γ₁ the intrinsic (absorption,radiation etc.) loss rate, while κ_(11-e) is the frequency shift inducedonto 1 due to the presence of e and Γ_(1-e) is the extrinsic due to e(absorption inside e, scattering from e etc.) loss rate. The first-orderPT model is valid only for small perturbations. Nevertheless, theparameters κ_(11-e), Γ_(1-e) are well defined, even outside that regime,if a₁ is taken to be the amplitude of the exact perturbed mode. Notealso that interference effects between the radiation field of theinitial resonant-object mode and the field scattered off the extraneousobject can for strong scattering (e.g. off metallic objects) result intotal radiation-Γ_(1-e)'s that are smaller than the initial radiation-Γ₁(namely Γ_(1-e) is negative).

The frequency shift is a problem that can be “fixed” by applying to oneor more resonant objects a feedback mechanism that corrects itsfrequency. For example, referring to FIG. 8 a, in some embodiments eachresonant object is provided with an oscillator at fixed frequency and amonitor which determines the frequency of the object. Both theoscillator and the monitor are coupled to a frequency adjuster which canadjust the frequency of the resonant object by, for example, adjustingthe geometric properties of the object (e.g. the height of aself-resonant coil, the capacitor plate spacing of a capacitively-loadedloop or coil, the dimensions of the inductor of an inductively-loadedrod, the shape of a dielectric disc, etc.) or changing the position of anon-resonant object in the vicinity of the resonant object. Thefrequency adjuster determines the difference between the fixed frequencyand the object frequency and acts to bring the object frequency intoalignment with the fixed frequency. This technique assures that allresonant objects operate at the same fixed frequency, even in thepresence of extraneous objects.

As another example, referring to FIG. 8 b, in some embodiments, duringenergy transfer from a source object to a device object, the deviceobject provides energy to a load, and an efficiency monitor measures theefficiency of the transfer. A frequency adjuster coupled to the load andthe efficiency monitor acts to adjust the frequency of the object tomaximize the transfer efficiency.

In various embodiments, other frequency adjusting schemes may be usedwhich rely on information exchange between the resonant objects. Forexample, the frequency of a source object can be monitored andtransmitted to a device object, which is in turn synched to thisfrequency using frequency adjusters as described above. In otherembodiments the frequency of a single clock may be transmitted tomultiple devices, and each device then synched to that frequency.

Unlike the frequency shift, the extrinsic loss can be detrimental to thefunctionality of the energy-transfer scheme, because it is difficult toremedy, so the total loss rate Γ_(1[e])=Γ₁Γ_(1-e) (and the correspondingfigure-of-merit κ_([e])/√{square root over (Γ_(1[e])Γ_(2[e]))}, whereκ_([e]) the perturbed coupling rate) should be quantified. Inembodiments using primarily magnetic resonances, the influence ofextraneous objects on the resonances is nearly absent. The reason isthat, in the quasi-static regime of operation (r<<λ) that we areconsidering, the near field in the air region surrounding the resonatoris predominantly magnetic (e.g. for coils with h<<2r most of theelectric field is localized within the self-capacitance of the coil orthe externally loading capacitor), therefore extraneous non-conductingobjects e that could interact with this field and act as a perturbationto the resonance are those having significant magnetic properties(magnetic permeability Re{μ}>1 or magnetic loss Im{μ}>0). Since almostall every-day non-conducting materials are non-magnetic but justdielectric, they respond to magnetic fields in the same way as freespace, and thus will not disturb the resonance of the resonator.Extraneous conducting materials can however lead to some extrinsiclosses due to the eddy currents induced on their surface.

As noted above, an extremely important implication of this fact relatesto safety considerations for human beings. Humans are also non-magneticand can sustain strong magnetic fields without undergoing any risk. Atypical example, where magnetic fields B˜1T are safely used on humans,is the Magnetic Resonance Imaging (MRI) technique for medical testing.In contrast, the magnetic near-field required in typical embodiments inorder to provide a few Watts of power to devices is only B˜10⁻⁴T, whichis actually comparable to the magnitude of the Earth's magnetic field.Since, as explained above, a strong electric near-field is also notpresent and the radiation produced from this non-radiative scheme isminimal, it is reasonable to expect that our proposed energy-transfermethod should be safe for living organisms.

One can, for example, estimate the degree to which the resonant systemof a capacitively-loaded conducting-wire coil has mostly magnetic energystored in the space surrounding it. If one ignores the fringing electricfield from the capacitor, the electric and magnetic energy densities inthe space surrounding the coil come just from the electric and magneticfield produced by the current in the wire; note that in the far field,these two energy densities must be equal, as is always the case forradiative fields. By using the results for the fields produced by asubwavelength (r<<λ) current loop (magnetic dipole) with h=0, we cancalculate the ratio of electric to magnetic energy densities, as afunction of distance D_(p) from the center of the loop (in the limitr<<D_(p)) and the angle θ with respect to the loop axis:

$\begin{matrix}{{{\frac{u_{e}(x)}{u_{m}(x)} = {\frac{ɛ_{o}{{E(x)}}^{2}}{\mu_{o}{{H(x)}}^{2}} = \frac{\left( {1 + \frac{1}{x^{2}}} \right)\sin^{2}\theta}{\begin{matrix}{{\left( {\frac{1}{x^{2}} + \frac{1}{x^{4}}} \right)4\; \cos^{2}\theta} +} \\{\left( {1 - \frac{1}{x^{2}} + \frac{1}{x^{4}}} \right)\sin^{2}\theta}\end{matrix}}}};}{{x = {\left. {2\; \pi \frac{D_{p}}{\lambda}}\Rightarrow\frac{∯\limits_{S_{p}}{{u_{e}(x)}{S}}}{∯\limits_{S_{p}}{{u_{m}(x)}{S}}} \right. = \frac{1 + \frac{1}{x^{2}}}{1 + \frac{1}{x^{2}} + \frac{3}{x^{4}}}}};}{{x = {2\; \pi \frac{D_{p}}{\lambda}}},}} & (12)\end{matrix}$

where the second line is the ratio of averages over all angles byintegrating the electric and magnetic energy densities over the surfaceof a sphere of radius D_(p). From Eq. (12) it is obvious that indeed forall angles in the near field (x<<1) the magnetic energy density isdominant, while in the far field (x>>1) they are equal as they shouldbe. Also, the preferred positioning of the loop is such that objectswhich may interfere with its resonance lie close to its axis (θ=0),where there is no electric field. For example, using the systemsdescribed in Table 4, we can estimate from Eq. (12) that for the loop ofr=30 cm at a distance D_(p)=10r=3 m the ratio of average electric toaverage magnetic energy density would be ˜12% and at D_(p)=3r=90 cm itwould be ˜1%, and for the loop of r=10 cm at a distance D_(p)=10r=1 mthe ratio would be ˜33% and at D_(p)=3r=30 cm it would be ˜2.5%. Atcloser distances this ratio is even smaller and thus the energy ispredominantly magnetic in the near field, while in the radiative farfield, where they are necessarily of the same order (ratio→1), both arevery small, because the fields have significantly decayed, ascapacitively-loaded coil systems are designed to radiate very little.Therefore, this is the criterion that qualifies this class of resonantsystem as a magnetic resonant system.

To provide an estimate of the effect of extraneous objects on theresonance of a capacitively-loaded loop including the capacitor fringingelectric field, we use the perturbation theory formula, stated earlier,Γ₁₋ c ^(abs)=ω₁/4·∫d³r Im{∈_(e)(r)}|E₁(r)|²/U with the computationalFEFD results for the field of an example like the one shown in the plotof FIG. 5 and with a rectangular object of dimensions 30 cm×30 cm×1.5 mand permittivity ∈=49+16i (consistent with human muscles) residingbetween the loops and almost standing on top of one capacitor (˜3 cmaway from it) and find Q_(c-h) ^(abs)˜10⁵ and for ˜10 cm away Q_(c-h)^(abs)5·10⁵. Thus, for ordinary distances (˜1 m) and placements (notimmediately on top of the capacitor) or for most ordinary extraneousobjects e of much smaller loss-tangent, we conclude that it is indeedfair to say that Q_(c-e) ^(abs)→∞. The only perturbation that isexpected to affect these resonances is a close proximity of largemetallic structures.

Self-resonant coils are more sensitive than capacitively-loaded coils,since for the former the electric field extends over a much largerregion in space (the entire coil) rather than for the latter (justinside the capacitor). On the other hand, self-resonant coils are simpleto make and can withstand much larger voltages than most lumpedcapacitors.

In general, different embodiments of resonant systems have differentdegree of sensitivity to external perturbations, and the resonant systemof choice depends on the particular application at hand, and howimportant matters of sensitivity or safety are for that application. Forexample, for a medical implantable device (such as a wirelessly poweredartificial heart) the electric field extent must be minimized to thehighest degree possible to protect the tissue surrounding the device. Insuch cases where sensitivity to external objects or safety is important,one should design the resonant systems so that the ratio of electric tomagnetic energy density u_(e)/u_(m) is reduced or minimized at most ofthe desired (according to the application) points in the surroundingspace.

In embodiments using resonances that are not primarily magnetic, theinfluence of extraneous objects may be of concern. For example, fordielectric disks, small, low-index, low-material-loss or far-away strayobjects will induce small scattering and absorption. In such cases ofsmall perturbations these extrinsic loss mechanisms can be quantifiedusing respectively the analytical first-order perturbation theoryformulas

Γ₁₋ e ^(rad)=ω₁ ∫d ³ rRe{∈ _(e)(r)}|E ₁(r)|² /U

and

Γ₁₋ e ^(abs)=ω₁/4·∫d ³ rIm{∈ _(e)(r)}|E ₁(r)|² /U

where U=1/2∫d³r∈(r)|E₁(r)|² is the total resonant electromagnetic energyof the unperturbed mode. As one can see, both of these losses depend onthe square of the resonant electric field tails E₁ at the site of theextraneous object. In contrast, the coupling rate from object 1 toanother resonant object 2 is, as stated earlier,

κ=ω₁/2·d ³ r∈ ₂(r)E ₂*(r)E ₁(r)/∫d ³ r∈(r)|E ₁(r)|²

and depends linearly on the field tails E₁ of 1 inside 2. Thisdifference in scaling gives us confidence that, for, for example,exponentially small field tails, coupling to other resonant objectsshould be much faster than all extrinsic loss rates (κ>>Γ_(1-e)), atleast for small perturbations, and thus the energy-transfer scheme isexpected to be sturdy for this class of resonant dielectric disks.However, we also want to examine certain possible situations whereextraneous objects cause perturbations too strong to analyze using theabove first-order perturbation theory approach. For example, we place adielectric disk c close to another off-resonance object of large Re{∈},Im{∈} and of same size but different shape (such as a human being h), asshown in FIG. 9 a, and a roughened surface of large extent but of smallRe{∈}, Im{∈} (such as a wall w), as shown in FIG. 9 b. For distancesD_(h/w)/r=10⁻³ between the disk-center and the “human”-center or “wall”,the numerical FDFD simulation results presented in FIGS. 9 a and 9 bsuggest that, the disk resonance seems to be fairly robust, since it isnot detrimentally disturbed by the presence of extraneous objects, withthe exception of the ver close proximity of high-loss objects. Toexamine the influence of large perturbations on an entireenergy-transfer system we consider two resonant disks in the closepresence of both a “human” and a “wall”. Comparing Table 8 to the tablein FIG. 9 c, the numerical FDFD simulations show that the systemperformance deteriorates from κ/Γ_(c˜)1-50 to κ[hw]/Γ_(c[hw])˜0.5-10i.e. only by acceptably small amounts.

Inductively-loaded conducting rods may also be more sensitive thancapacitively-loaded coils, since they rely on the electric field toachieve the coupling.

System Efficiency

In general, another important factor for any energy transfer scheme isthe transfer efficiency. Consider again the combined system of aresonant source s and device d in the presence of a set of extraneousobjects e. The efficiency of this resonance-based energy-transfer schememay be determined, when energy is being drained from the device at rateΓ_(work) for use into operational work. The coupled-mode-theory equationfor the device field-amplitude is

$\begin{matrix}{{\frac{a_{d}}{t} = {{{- {\left( {\omega - {\; \Gamma_{d{\lbrack e\rbrack}}}} \right)}}a_{d}} + {\; \kappa_{\lbrack e\rbrack}a_{s}} - {\Gamma_{work}a_{d}}}},} & (13)\end{matrix}$

where Γ_(d[e])=Γ_(d[e]) ^(rad)+Γ_(d[e]) ^(abs)=Γ_(d[e]) ^(rad)+(Γ_(d)^(abs)+Γ_(d-e) ^(abs)) is the net perturbed-device loss rate, andsimilarly we define Γ_(s[e]) for the perturbed-source. Differenttemporal schemes can be used to extract power from the device (e.g.steady-state continuous-wave drainage, instantaneous drainage atperiodic times and so on) and their efficiencies exhibit differentdependence on the combined system parameters. For simplicity, we assumesteady state, such that the field amplitude inside the source ismaintained constant, namely a_(s)(t)=A_(s)e^(−iωt), so then the fieldamplitude inside the device is a_(d)(t)=A_(d)e^(−iωt) withA_(d)/A_(s)=iκ_([e])/(Γ_(d[e])+Γ_(work)). The various time-averagedpowers of interest are then: the useful extracted power isP_(work)=2Γ_(work)|A_(d)|², the radiated (including scattered) power isP_(rad)=2Γ_(s[e]) ^(rad)|A_(s)|²+2Γ_(s[e]) ^(rad)|A_(d)|², the powerabsorbed at the source/device is P_(s/d)=2Γ_(s/d) ^(abs)|A_(s/d)|², andat the extraneous objects P_(e)=2Γ_(s-e) ^(abs)|A_(s)|²+2sΓ_(d-e)^(abs)|A_(d)|². From energy conservation, the total time-averaged powerentering the system is P_(total)=P_(work)+P_(rad)+P_(s)+P_(d)+P_(e).Note that the reactive powers, which are usually present in a system andcirculate stored energy around it, cancel at resonance (which can beproven for example in electromagnetism from Poynting's Theorem) and donot influence the power-balance calculations. The working efficiency isthen:

$\begin{matrix}{{{\eta_{work} \equiv \frac{P_{work}}{P_{total}}} = \frac{1}{1 + {\frac{\Gamma_{d{\lbrack e\rbrack}}}{\Gamma_{work}} \cdot \left\lbrack {1 + {\frac{1}{{fom}_{\lbrack e\rbrack}^{2}}\left( {1 + \frac{\Gamma_{work}}{\Gamma_{d{\lbrack e\rbrack}}}} \right)^{2}}} \right\rbrack}}},} & (14)\end{matrix}$

where fom_([e])=κ_([e])/√{square root over (Γ_(s[e])Γ_(d[e]))} is thedistance-dependent figure-of-merit of the perturbed resonantenergy-exchange system. To derive Eq. (14), we have assumed that therate Γ_(supply), at which the power supply is feeding energy to theresonant source, is Γ_(supply)=Γ_(s[e])+κ²/(Γ_(d[e])+Γ_(work)), suchthat there are zero reflections of the fed power P_(total) back into thepower supply.

[Example: Capacitively-Loaded Conducting Loops]

Referring to FIG. 10, to rederive and express this formula (14) in termsof the parameters which are more directly accessible from particularresonant objects, e.g. the capacitively-loaded conducting loops, one canconsider the following circuit-model of the system, where theinductances L_(s), L_(d) represent the source and device loopsrespectively, R_(s), R_(d) their respective losses, and C_(s), C_(d) arethe required corresponding capacitances to achieve for both resonance atfrequency ω. A voltage generator V_(g) is considered to be connected tothe source and a work (load) resistance R_(w) to the device. The mutualinductance is denoted by M.

Then from the source circuit at resonance (ωL_(s)=1/ωC_(s)):

V _(g) =I _(s) R _(s) −jωMI _(d)

½V _(g) *I _(s)=½|I _(s)|² R _(s)+½jωMI _(d) *I _(s),

and from the device circuit at resonance (ωL_(d)=1/ωC_(d)):

0=I _(d)(R _(d) +R _(w))−jωMI _(s)

jωMI _(s) =I _(d)(R _(d) +R _(w))

So by substituting the second to the first:

½V _(g) *I _(s)=½|I _(s)|² R _(s)+½|I _(d)|²(R _(d) +R _(w)).

Now we take the real part (time-averaged powers) to find the efficiency:

$\begin{matrix}{P_{g} \equiv {{Re}\left\{ {\frac{1}{2}V_{g}^{*}I_{s}} \right\}}} \\{= \left. {P_{s} + P_{d} + P_{w}}\Rightarrow\eta_{work} \right.} \\{\equiv \frac{P_{w}}{P_{tot}}} \\{= {\frac{R_{w}}{{\frac{I_{s}}{I_{d}}}^{2}{{\cdot R_{s}} + R_{d} + R_{w}}}.}}\end{matrix}$

Namely,

${\eta_{work} = \frac{R_{w}}{{\frac{\left( {R_{d} + R_{w}} \right)^{2}}{\left( {\omega \; M} \right)^{2}} \cdot R_{s}} + R_{d} + R_{w}}},$

which with Γ_(work)=R_(w)/2L_(d), Γ_(d)=R_(d)/2L_(d),Γ_(s)=R_(s)/2L_(s), and κ=ωM/2√{square root over (Γ_(s)Γ_(d))}, becomesthe general Eq. (14). [End of Example]

From Eq. (14) one can find that the efficiency is optimized in terms ofthe chosen work-drainage rate, when this is chosen to beΓ_(work)/Γ_(d[e])=Γ_(supply)/Γ_(s[e])=√{square root over (1+fom_([e])²)}>1. Then, η_(work) is a function of the fom_([e]) parameter only asshown in FIG. 11 with a solid black line. One can see that theefficiency of the system is η>17% for fom_([e])>1, large enough forpractical applications. Thus, the efficiency can be further increasedtowards 100% by optimizing fom_([e]) as described above. The ratio ofconversion into radiation loss depends also on the other systemparameters, and is plotted in FIG. 11 for the conducting loops withvalues for their parameters within the ranges determined earlier.

For example, consider the capacitively-loaded coil embodiments describedin Table 4, with coupling distance D/r=7, a “human” extraneous object atdistance D_(h) from the source, and that P_(work)=10 W must be deliveredto the load. Then, we have (based on FIG. 11) Q_(s[h]) ^(rad)=Q_(d[h])^(rad)˜10⁴, Q_(s) ^(abs)=Q_(d) ^(abs)˜10³, Q_(κ)˜500, and Q_(d-h)^(abs)→∞, Q_(s-h) ^(abs)˜10⁵ at D_(h)˜3 cm and Q_(s-h) ^(abs)˜5·10⁵ atD_(h)˜10 cm. Therefore fom_([h])˜2, so we find η≈38%, P_(rad)≈1.5 W,P_(s)≈11 W, P_(d)≈4 W, and most importantly η_(h)≈0.4%, P_(h)=0.1 W atD_(h)·3 cm and η_(h)≈0.1%, P_(h)=0.02 W at D_(h)˜10 cm.

Overall System Performance

In many cases, the dimensions of the resonant objects will be set by theparticular application at hand. For example, when this application ispowering a laptop or a cell-phone, the device resonant object cannothave dimensions larger that those of the laptop or cell-phonerespectively. In particular, for a system of two loops of specifieddimensions, in terms of loop radii r_(s,d) and wire radii a_(s,d), theindependent parameters left to adjust for the system optimization are:the number of turns N_(s,d), the frequency f, the work-extraction rate(load resistance) Γ_(work) and the power-supply feeding rate Γ_(supply).

In general, in various embodiments, the primary dependent variable thatone wants to increase or optimize is the overall efficiency η. However,other important variables need to be taken into consideration uponsystem design. For example, in embodiments featuring capacitively-loadedcoils, the design may be constrained by, for example, the currentsflowing inside the wires I_(s,d) and the voltages across the capacitorsV_(s,d). These limitations can be important because for ˜Watt powerapplications the values for these parameters can be too large for thewires or the capacitors respectively to handle. Furthermore, the totalloaded Q_(tot)=ωL_(d)/(R_(d)+R_(w)) of the device is a quantity thatshould be preferably small, because to match the source and deviceresonant frequencies to within their Q's, when those are very large, canbe challenging experimentally and more sensitive to slight variations.Lastly, the radiated powers P_(rad,s,d) should be minimized for safetyconcerns, even though, in general, for a magnetic, non-radiative schemethey are already typically small.

In the following, we examine then the effects of each one of theindependent variables on the dependent ones. We define a new variable wpto express the work-drainage rate for some particular value of fom_([e])through Γ_(work)/Γ_(d[e])=√{square root over (1+wp·fom_([e]) ²)}. Then,in some embodiments, values which impact the choice of this rate are:Γ_(work)/Γ_(d[e])=1

wp=0 to minimize the required energy stored in the source (and thereforeI_(s) and V_(s)), Γ_(work)/Γ_(d[e])=√{square root over (1+fom_([e])²)}>1

wp=1 to increase the efficiency, as seen earlier, orΓ_(work)/Γ_(d[e])>>1

wp>>1 to decrease the required energy stored in the device (andtherefore I_(d) and V_(d)) and to decrease or minimizeQ_(tot)=ωL_(d)/(R_(d)+R_(w))=ω/[2(Γ_(d)+Γ_(work))]. Similar is theimpact of the choice of the power supply feeding rate Γ_(supply), withthe roles of the source and the device reversed.

Increasing N_(s) and N_(d) increases κ/√{square root over (Γ_(s)Γ_(d))}and thus efficiency significantly, as seen before, and also decreasesthe currents I_(s) and I_(d), because the inductance of the loopsincreases, and thus the energy U_(s,d)=½L_(s,d)|I_(s,d)|² required forgiven output power P_(work) can be achieved with smaller currents.However, increasing N_(d) increases Q_(tot), P_(rad,d) and the voltageacross the device capacitance V_(d), which unfortunately ends up being,in typical embodiments one of the greatest limiting factors of thesystem. To explain this, note that it is the electric field that reallyinduces breakdown of the capacitor material (e.g. 3 kV/mm for air) andnot the voltage, and that for the desired (close to the optimal)operational frequency, the increased inductance L_(d) implies reducedrequired capacitance C_(d), which could be achieved in principle, for acapacitively-loaded device coil by increasing the spacing of the devicecapacitor plates d_(d) and for a self-resonant coil by increasingthrough h_(d) the spacing of adjacent turns, resulting in an electricfield (≈V_(d)/d_(d) for the former case) that actually decreases withN_(d); however, one cannot in reality increase d_(d) or h_(d) too much,because then the undesired capacitance fringing electric fields wouldbecome very large and/or the size of the coil might become too large;and, in any case, for certain applications extremely high voltages arenot desired. A similar increasing behavior is observed for the sourceP_(rad,s) and V_(s) upon increasing N_(s). As a conclusion, the numberof turns N_(s) and N_(d) have to be chosen the largest possible (forefficiency) that allow for reasonable voltages, fringing electric fieldsand physical sizes.

With respect to frequency, again, there is an optimal one forefficiency, and Q_(tot) is approximately maximum, close to that optimalfrequency. For lower frequencies the currents get worse (larger) but thevoltages and radiated powers get better (smaller). Usually, one shouldpick either the optimal frequency or somewhat lower.

One way to decide on an operating regime for the system is based on agraphical method. In FIG. 12, for two loops of r_(s)=25 cm, r_(d)=15 cm,h_(s)=h_(d)=0, a_(s)=a_(d)=3 mm and distance D=2 m between them, we plotall the above dependent variables (currents, voltages and radiatedpowers normalized to 1 Watt of output power) in terms of frequency andN_(d), given some choice for wp and N_(s). The Figure depicts all of thedependencies explained above. We can also make a contour plot of thedependent variables as functions of both frequency and wp but for bothN_(s) and N_(d) fixed. The results are shown in FIG. 13 for the sameloop dimensions and distance. For example, a reasonable choice ofparameters for the system of two loops with the dimensions given aboveare: N_(s)=2, N_(d)=6, f=10 MHz and wp=10, which gives the followingperformance characteristics: η_(work)=20.6%, Q_(tot)=1264, I_(s)=7.2A,I_(d)=1.4A, V_(s)=2.55 kV, V_(d)=2.30 kV, P_(rad,s)=0.15 W,P_(rad,d)=0.006 W Note that the results in FIGS. 12 and 13, and the justabove calculated performance characteristics are made using theanalytical formulas provided above, so they are expected to be lessaccurate for large values of N_(s), N_(d), still they give a goodestimate of the scalings and the orders of magnitude.

Finally, one could additionally optimize for the source dimensions,since usually only the device dimensions are limited, as discussedearlier. Namely, one can add r_(s) and a_(s) in the set of independentvariables and optimize with respect to these too for all the dependentvariables of the problem (we saw how to do this only for efficiencyearlier). Such an optimization would lead to improved results.

Experimental Results

An experimental realization of an embodiment of the above describedscheme for wireless energy transfer consists of two self-resonant coilsof the type described above, one of which (the source coil) is coupledinductively to an oscillating circuit, and the second (the device coil)is coupled inductively to a resistive load, as shown schematically inFIG. 14. Referring to FIG. 14, A is a single copper loop of radius 25 cmthat is part of the driving circuit, which outputs a sine wave withfrequency 9.9 MHz. s and d are respectively the source and device coilsreferred to in the text. B is a loop of wire attached to the load(“light-bulb”). The various κ's represent direct couplings between theobjects. The angle between coil d and the loop A is adjusted so thattheir direct coupling is zero, while coils s and d are alignedcoaxially. The direct coupling between B and A and between B and s isnegligible.

The parameters for the two identical helical coils built for theexperimental validation of the power transfer scheme were h=20 cm, a=3mm, r=30 cm, N=5.25. Both coils are made of copper. Due to imperfectionsin the construction, the spacing between loops of the helix is notuniform, and we have encapsulated the uncertainty about their uniformityby attributing a 10% (2 cm) uncertainty to h. The expected resonantfrequency given these dimensions is f₀=10.56±0.3 MHz, which is about 5%off from the measured resonance at around 9.90 MHz.

The theoretical Q for the loops is estimated to be ˜2500 (assumingperfect copper of resistivity ρ=1/σ=1.7×10⁻⁸ Ωm) but the measured valueis 950±50. We believe the discrepancy is mostly due to the effect of thelayer of poorly conducting copper oxide on the surface of the copperwire, to which the current is confined by the short skin depth (˜20 μm)at this frequency. We have therefore used the experimentally observed Q(and Γ₁=Γ₂=Γ=ω)/(2Q) derived from it) in all subsequent computations.

The coupling coefficient κ can be found experimentally by placing thetwo self-resonant coils (fine-tuned, by slightly adjusting h, to thesame resonant frequency when isolated) a distance D apart and measuringthe splitting in the frequencies of the two resonant modes in thetransmission spectrum. According to coupled-mode theory, the splittingin the transmission spectrum should be Δω=2√{square root over (κ²−Γ²)}.The comparison between experimental and theoretical results as afunction of distance when the two the coils are aligned coaxially isshown in FIG. 15.

FIG. 16 shows a comparison of experimental and theoretical values forthe parameter κ/Γ as a function of the separation between the two coils.The theory values are obtained by using the theoretically obtained K andthe experimentally measured Γ. The shaded area represents the spread inthe theoretical κ/Γ due to the ˜5% uncertainty in Q.

As noted above, the maximum theoretical efficiency depends only on theparameter κ/√{square root over (Γ₁Γ₂)}=κ/Γ, plotted as a function ofdistance in FIG. 17. The coupling to loss ratio κ/Γ is greater than 1even for D=2.4 m (eight times the radius of the coils), thus the systemis in the strongly-coupled regime throughout the entire range ofdistances probed.

The power supply circuit was a standard Colpitts oscillator coupledinductively to the source coil by means of a single loop of copper wire25 cm in radius (see FIG. 14). The load consisted of a previouslycalibrated light-bulb, and was attached to its own loop of insulatedwire, which was in turn placed in proximity of the device coil andinductively coupled to it. Thus, by varying the distance between thelight-bulb and the device coil, the parameter Γ_(work)/Γ was adjusted sothat it matched its optimal value, given theoretically by √{square rootover (1+κ²/(Γ₁Γ₂))}. Because of its inductive nature, the loop connectedto the light-bulb added a small reactive component to Γ_(work) which wascompensated for by slightly retuning the coil. The work extracted wasdetermined by adjusting the power going into the Colpitts oscillatoruntil the light-bulb at the load was at its full nominal brightness.

In order to isolate the efficiency of the transfer taking placespecifically between the source coil and the load, we measured thecurrent at the mid-point of each of the self-resonant coils with acurrent-probe (which was not found to lower the Q of the coilsnoticeably.) This gave a measurement of the current parameters I₁ and I₂defined above. The power dissipated in each coil was then computed fromP_(1,2)=ΓL|I_(1,2)|², and the efficiency was directly obtained fromη=P_(work)/(P₁+P₂+P_(work)). To ensure that the experimental setup waswell described by a two-object coupled-mode theory model, we positionedthe device coil such that its direct coupling to the copper loopattached to the Colpitts oscillator was zero. The experimental resultsare shown in FIG. 17, along with the theoretical prediction for maximumefficiency, given by Eq. (14).

Using this embodiment, we were able to transfer significant amounts ofpower using this setup, fully lighting up a 60 W light-bulb fromdistances more than 2 m away, for example. As an additional test, wealso measured the total power going into the driving circuit. Theefficiency of the wireless transfer itself was hard to estimate in thisway, however, as the efficiency of the Colpitts oscillator itself is notprecisely known, although it is expected to be far from 100%.Nevertheless, this gave an overly conservative lower bound on theefficiency. When transferring 60 W to the load over a distance of 2 m,for example, the power flowing into the driving circuit was 400 W. Thisyields an overall wall-to-load efficiency of ˜15%, which is reasonablegiven the expected ˜40% efficiency for the wireless power transfer atthat distance and the low efficiency of the driving circuit.

From the theoretical treatment above, we see that in typical embodimentsit is important that the coils be on resonance for the power transfer tobe practical. We found experimentally that the power transmitted to theload dropped sharply as one of the coils was detuned from resonance. Fora fractional detuning Δf/f₀ of a few times the inverse loaded Q, theinduced current in the device coil was indistinguishable from noise.

The power transfer was not found to be visibly affected as humans andvarious everyday objects, such as metallic and wooden furniture, as wellas electronic devices large and small, were placed between the twocoils, even when they drastically obstructed the line of sight betweensource and device. External objects were found to have an effect onlywhen they were closer than 10 cm from either one of the coils. Whilesome materials (such as aluminum foil, styrofoam and humans) mostly justshifted the resonant frequency, which could in principle be easilycorrected with a feedback circuit of the type described earlier, others(cardboard, wood, and PVC) lowered Q when placed closer than a fewcentimeters from the coil, thereby lowering the efficiency of thetransfer.

We believe that this method of power transfer should be safe for humans.When transferring 60 W (more than enough to power a laptop computer)across 2 m, we estimated that the magnitude of the magnetic fieldgenerated is much weaker than the Earth's magnetic field for alldistances except for less than about 1 cm away from the wires in thecoil, an indication of the safety of the scheme even after long-termuse. The power radiated for these parameters was ˜5 W, which is roughlyan order of magnitude higher than cell phones but could be drasticallyreduced, as discussed below.

Although the two coils are currently of identical dimensions, it ispossible to make the device coil small enough to fit into portabledevices without decreasing the efficiency. One could, for instance,maintain the product of the characteristic sizes of the source anddevice coils constant.

These experiments demonstrated experimentally a system for powertransfer over medium range distances, and found that the experimentalresults match theory well in multiple independent and mutuallyconsistent tests.

We believe that the efficiency of the scheme and the distances coveredcould be appreciably improved by silver-plating the coils, which shouldincrease their Q, or by working with more elaborate geometries for theresonant objects. Nevertheless, the performance characteristics of thesystem presented here are already at levels where they could be usefulin practical applications.

Applications

In conclusion, we have described several embodiments of aresonance-based scheme for wireless non-radiative energy transfer.Although our consideration has been for a static geometry (namely κ andΓ_(e) were independent of time), all the results can be applied directlyfor the dynamic geometries of mobile objects, since the energy-transfertime κ⁻¹ (˜1 μs-1 ms for microwave applications) is much shorter thanany timescale associated with motions of macroscopic objects. Analysesof very simple implementation geometries provide encouraging performancecharacteristics and further improvement is expected with serious designoptimization. Thus the proposed mechanism is promising for many modernapplications.

For example, in the macroscopic world, this scheme could potentially beused to deliver power to for example, robots and/or computers in afactory room, or electric buses on a highway. In some embodimentssource-object could be an elongated “pipe” running above the highway, oralong the ceiling.

Some embodiments of the wireless transfer scheme can provide energy topower or charge devices that are difficult or impossible to reach usingwires or other techniques. For example some embodiments may providepower to implanted medical devices (e.g. artificial hearts, pacemakers,medicine delivery pumps, etc.) or buried underground sensors.

In the microscopic world, where much smaller wavelengths would be usedand smaller powers are needed, one could use it to implement opticalinter-connects for CMOS electronics, or to transfer energy to autonomousnano-objects (e.g. MEMS or nano-robots) without worrying much about therelative alignment between the sources and the devices. Furthermore, therange of applicability could be extended to acoustic systems, where thesource and device are connected via a common condensed-matter object.

In some embodiments, the techniques described above can providenon-radiative wireless transfer of information using the localized nearfields of resonant object. Such schemes provide increased securitybecause no information is radiated into the far-field, and are wellsuited for mid-range communication of highly sensitive information.

A number of embodiments of the invention have been described.Nevertheless, it will be understood that various modifications may bemade without departing from the spirit and scope of the invention.

1. An apparatus for use in wireless energy transfer, the apparatuscomprising: a first resonator structure configured for energy transferwith a second resonator structure, over a distance D larger than acharacteristic size L₁ of said first resonator structure and larger thana characteristic size L₂ of said second resonator structure, wherein theenergy transfer has a rate K and is mediated by evanescent-tail couplingof a resonant field of the first resonator structure and a resonantfield of the second resonator structure, wherein said resonant field ofthe first resonator structure has a resonance angular frequency ω₁, aresonance frequency-width Γ₁, and a resonance quality factor Q₁=ω₁/2Γ₁at least larger than 300, and said resonant field of the secondresonator structure has a resonance angular frequency ω₂, a resonancefrequency-width Γ₂, and a resonance quality factor Q₂=ω₂/2Γ₂ at leastlarger than 300, wherein the absolute value of the difference of saidangular frequencies ω₁ and ω₂ is smaller than the broader of saidresonant widths Γ₁ and Γ₂, and the quantity κ/√{square root over (Γ₁Γ₂)}is at least larger than
 20. 2. The apparatus of claim 1, wherein Q, isat least larger than 1000, Q₂ is at least larger than 1000, and thequantity κ/√{square root over (Γ₁Γ₂)} is at least larger than
 50. 3. Theapparatus of claim 1, wherein D/L₂ is at least larger than 5, and thequantity κ/√{square root over (Γ₁Γ₂)} is at least larger than
 2. 4. Theapparatus of claim 1, wherein D/L₂ is at least larger than 5, Q, is atleast larger than 1000, Q₂ is at least larger than 1000, and thequantity κ/√{square root over (Γ₁Γ₂)} is at least larger than
 5. 5. Theapparatus of claim 1, wherein D/L₂ is at least larger than 10, and thequantity κ/√{square root over (Γ₁Γ₂)} is at least larger than 0.4. 6.The apparatus of claim 1, wherein D/L₂ is at least larger than 10, Q, isat least larger than 1000, Q₂ is at least larger than 1000, and thequantity κ/√{square root over (Γ₁Γ₂)} is at least larger than
 1. 7. Theapparatus of claim 1, wherein the quantity κ/√{square root over (Γ₁Γ₂)}is maximized at an angular frequency {tilde over (ω)} with a frequencywidth {tilde over (Γ)}, and the absolute value of the difference of theangular frequencies ω₁ and {tilde over (ω)} is smaller than the width{tilde over (Γ)}, and the absolute value of the difference of theangular frequencies ω₂ and {tilde over (ω)} is smaller than the width{tilde over (Γ)}.
 8. The apparatus of claim 1, further comprising thesecond resonator structure.
 9. The apparatus of claim 1, wherein adevice coupled to the first or second resonator structure with acoupling rate Γ_(work) receives from the resonator structure, to whichit is coupled, a usable power P_(work) at least larger than 0.03 Watt.10. The apparatus of claim 9, wherein P_(work) is at least larger than 1Watt.
 11. The apparatus of claim 9, wherein P_(work) is at least largerthan 30 Watt.
 12. The apparatus of claim 9, wherein, if the device iscoupled to the first resonator, the quantity[(Γ_(work)/Γ₁)²−1]/(κ/√{square root over (Γ₁Γ₂)})² is at least largerthan 0.5 and smaller than 2, and if the device is coupled to the secondresonator, the quantity [(Γ_(work)/Γ₂)²−1]/(κ/√{square root over(Γ₁Γ₂)})² is at least larger than 0.5 and smaller than
 2. 13. Theapparatus of claim 9, wherein the device comprises at least one of arobot, a mobile electronic device, a computer, a sensor, a vehicle, amedical device configured to be implanted in a patient, a televisionscreen, a wireless keyboard, and a wireless mouse.
 14. The apparatus ofclaim 9, further comprising the device.
 15. The apparatus of claim 9,wherein, during operation, a power supply coupled to the first or secondresonator structure with a coupling rate Γ_(supply) drives the resonatorstructure, to which it is coupled, at a frequency f and supplies powerP_(total), wherein the absolute value of the difference of the angularfrequencies ω=2πf and ω₁ is smaller than the resonant width Γ₁, and theabsolute value of the difference of the angular frequencies ω=2πf and ω₂is smaller than the resonant width Γ₂.
 16. The apparatus of claim 15,wherein, if the power supply is coupled to the first resonator, thequantity [(Γ_(supply)/Γ₁)²−1]/(κ/√{square root over (Γ₁Γ₂)})² is atleast larger than 0.5 and smaller than 2, and if the power supply iscoupled to the second resonator, the quantity[(Γ_(supply)/Γ₂)²−1]/(κ/√{square root over (Γ₁Γ₂)})² is at least largerthan 0.5 and smaller than
 2. 17. The apparatus of claim 15, furthercomprising the power supply.
 18. The apparatus of claim 15, wherein,during the energy exchange, power P_(rad) is radiated and the radiationloss η_(rad)=P_(rad)/P_(total) is less that 10%.
 19. The apparatus ofclaim 15, wherein, during the energy exchange, power P_(rad) is radiatedand the radiation loss η_(rad)=P_(rad)/P_(total), is less that 1%. 20.The apparatus of claim 1, wherein the resonant fields areelectromagnetic.
 21. The apparatus of claim 20, wherein f is at leastlarger than 100 kHz and smaller than 500 MHz.
 22. The apparatus of claim20, wherein f is at least larger than 1 MHz and smaller than 50 MHz. 23.The apparatus of claim 20, wherein f is within one of the frequencybands specially assigned for industrial, scientific, and medical (ISM)equipment.
 24. The apparatus of claim 20, wherein the resonant fieldsare primarily magnetic at a distance D_(p) from the center of theclosest resonant object, whose characteristic size is L_(R), and theratio of the average electric field energy density to the averagemagnetic field energy density is smaller than 0.1 for D_(p)/L_(R) lessthan
 3. 25. The apparatus of claim 20, wherein the resonant fields areprimarily magnetic at a distance D_(p) from the center of the closestresonant object, whose characteristic size is L_(R), and the ratio ofthe average electric field energy density to the average magnetic fieldenergy density is smaller than 0.01 for D_(p)/L_(R) less than
 3. 26. Theapparatus of claim 20, wherein at least one of the first and secondresonator structures comprises a self resonant coil of at least one of aconducting wire, a conducting Litz wire, and a conducting ribbon. 27.The apparatus of claim 26, wherein the resonator structure receivingenergy from the other resonator structure comprises a self resonant coilof conducting wire, wherein said coil is helical, the radius of saidcoil is about 10 cm, the height of said coil is about 3 cm, the numberof turns of said coil is about 6, and the radius of said conducting wireis about 2 mm.
 28. The apparatus of claim 26, wherein the resonatorstructure receiving energy from the other resonator structure comprisesa self resonant coil of conducting wire, wherein said coil is helical,the radius of said coil is about 30 cm, the height of said coil is about20 cm, the number of turns of said coil is about 5.25, and the radius ofsaid conducting wire is about 3 mm.
 29. The apparatus of claim 20,wherein at least one of the first and second resonator structurescomprises a capacitively loaded loop or coil of at least one of aconducting wire, a conducting Litz wire, and a conducting ribbon. 30.The apparatus of claim 29, wherein the resonator structure receivingenergy from the other resonator structure comprises a capacitivelyloaded loop of at least one of a conducting wire, a conducting Litzwire, and a conducting ribbon, wherein the characteristic size of saidloop is less than 10 cm and the width of said conducting wire or Litzwire or ribbon is less than 1 cm.
 31. The apparatus of claim 29, whereinthe resonator structure receiving energy from the other resonatorstructure comprises a capacitively loaded loop of at least one of aconducting wire, a conducting Litz wire, and a conducting ribbon,wherein the characteristic size of said loop is less than 30 cm and thewidth of said conducting wire or Litz wire or ribbon is less than 5 cm.32. The apparatus of claim 20, wherein at least one of the first andsecond resonator structures comprises an inductively loaded rod of atleast one of a conducting wire, a conducting Litz wire, and a conductingribbon.
 33. The apparatus of claim 32, wherein the resonator structurereceiving energy from the other resonator structure comprises aninductively loaded rod of at least one of a conducting wire, aconducting Litz wire, and a conducting ribbon, wherein thecharacteristic size of said rod is less than 10 cm and the width of saidconducting wire or Litz wire or ribbon is less than 1 cm.
 34. Theapparatus of claim 32, wherein the resonator structure receiving energyfrom the other resonator structure comprises an inductively loaded rodof at least one of a conducting wire, a conducting Litz wire, and aconducting ribbon, wherein the characteristic size of said rod is lessthan 30 cm and the width of said conducting wire or Litz wire or ribbonis less than 5 cm.
 35. The apparatus of claim 1, further comprising afeedback mechanism for maintaining the resonant frequency of one or moreof the resonant objects.
 36. The apparatus of claim 35, wherein thefeedback mechanism comprises an oscillator with a fixed frequency and isconfigured to adjust the resonant frequency of the one or more resonantobjects to be about equal to the fixed frequency.
 37. The apparatus ofclaim 35, where the feedback mechanism is configured to monitor anefficiency of the energy transfer, and adjust the resonant frequency ofthe one or more resonant objects to maximize the efficiency.
 38. Anapparatus for use in wireless energy transfer, the apparatus comprising:a first resonator structure configured for energy transfer with a secondresonator structure, over a distance D larger than a characteristicthickness T₁ of said first resonator structure and larger than acharacteristic size L₂ of said second resonator structure, wherein theenergy transfer has a rate K and is mediated by evanescent-tail couplingof a resonant field of the first resonator structure and a resonantfield of the second resonator structure, wherein said resonant field ofthe first resonator structure has a resonance angular frequency ω₁, aresonance frequency-width Γ₁, and a resonance quality factor Q₁=ω₁/2Γ₁at least larger than 300, and said resonant field of the secondresonator structure has a resonance angular frequency ω₂, a resonancefrequency-width Γ₂, and a resonance quality factor Q₂=ω₂/2Γ₂ at leastlarger than 300, wherein the absolute value of the difference of saidangular frequencies ω₁ and ω₂ is smaller than the broader of saidresonant widths Γ₁ and Γ₂, and the quantity κ/√{square root over (Γ₁Γ₂)}is at least larger than
 20. 39. The apparatus of claim 38, wherein Q₁ isat least larger than 1000, Q₂ is at least larger than 1000, and thequantity κ/√{square root over (Γ₁Γ₂)} is at least larger than
 50. 40.The apparatus of claim 38, wherein D/L₂ is at least larger than 5, andthe quantity κ/√{square root over (Γ₁Γ₂)} is at least larger than
 2. 41.The apparatus of claim 38, wherein D/L₂ is at least larger than 5, Q₁ isat least larger than 1000, Q₂ is at least larger than 1000, and thequantity κ/√{square root over (Γ₁Γ₂)} is at least larger than
 5. 42. Theapparatus of claim 38, wherein D/L₂ is at least larger than 10, and thequantity κ/√{square root over (Γ₁Γ₂)} is at least larger than 0.4. 43.The apparatus of claim 38, wherein D/L₂ is at least larger than 10, Q₁is at least larger than 1000, Q₂ is at least larger than 1000, and thequantity κ/√{square root over (Γ₁Γ₂)} is at least larger than
 1. 44. Theapparatus of claim 38, wherein the quantity κ/√{square root over (Γ₁Γ₂)}is maximized at an angular frequency {tilde over (ω)} with a frequencywidth {tilde over (Γ)}, and the absolute value of the difference of theangular frequencies ω₁ and {tilde over (ω)} is smaller than the width F,and the absolute value of the difference of the angular frequencies ω₂and {tilde over (ω)} is smaller than the width {tilde over (Γ)}.
 45. Theapparatus of claim 38, further comprising the second resonatorstructure.
 46. The apparatus of claim 38, wherein a device coupled tothe first or second resonator structure with a coupling rate Γ_(work)receives from the resonator structure, to which it is coupled, a usablepower P_(work) at least larger than 0.03 Watt.
 47. The apparatus ofclaim 46, wherein P_(work) is at least larger than 1 Watt.
 48. Theapparatus of claim 46, wherein P_(work) is at least larger than 30 Watt.49. The apparatus of claim 46, wherein, if the device is coupled to thefirst resonator, the quantity [(Γ_(work)/Γ₁)²−1]/(κ/√{square root over(Γ₁Γ₂)})² is at least larger than 0.5 and smaller than about 2, and ifthe device is coupled to the second resonator, the quantity[(F_(work)/Γ₂)²−1]/(κ/√{square root over (Γ₁Γ₂)})² is at least largerthan 0.5 and smaller than about
 2. 50. The apparatus of claim 46,wherein the device comprises at least one of a robot, a mobileelectronic device, a computer, a sensor, a vehicle, a medical deviceconfigured to be implanted in a patient, a television screen, a wirelesskeyboard, and a wireless mouse.
 51. The apparatus of claim 46, furthercomprising the device.
 52. The apparatus of claim 46, wherein, duringoperation, a power supply coupled to the first or second resonatorstructure with a coupling rate Γ_(supply) drives the resonatorstructure, to which it is coupled, at a frequency f and supplies powerP_(total), wherein the absolute value of the difference of the angularfrequencies ω=2πf and ω₁ is smaller than the resonant width Γ₁, and theabsolute value of the difference of the angular frequencies ω=2πf and ω₂is smaller than the resonant width Γ₂.
 53. The apparatus of claim 52,wherein, if the power supply is coupled to the first resonator, thequantity [(Γ_(supply)/Γ₁)²−1]/(κK/√{square root over (Γ₁Γ₂)})² is atleast larger than 0.5 and smaller than about 2, and if the power supplyis coupled to the second resonator, the quantity[(Γ_(supply)/Γ₂)²−1]/(κ/√{square root over (Γ₁Γ²)})² is at least largerthan 0.5 and smaller than about
 2. 54. The apparatus of claim 52,further comprising the power supply.
 55. The apparatus of claim 52,wherein, during the energy exchange, power P_(rad) is radiated and theradiation loss η_(rad)=P_(rad)/P_(total) is less that 10%.
 56. Theapparatus of claim 52, wherein, during the energy exchange, powerP_(rad) is radiated and the radiation loss η_(rad)=P_(rad)/P_(total), isless that 1%.
 57. The apparatus of claim 38, wherein the resonant fieldsare electromagnetic.
 58. The apparatus of claim 57, wherein f is atleast larger than 100 kHz and smaller than 500 MHz.
 59. The apparatus ofclaim 57, wherein f is at least larger than 1 MHz and smaller than 50MHz.
 60. The apparatus of claim 57, wherein f is within one of thefrequency bands specially assigned for industrial, scientific and,medical (ISM) equipment.
 61. The apparatus of claim 57, wherein theresonant fields are primarily magnetic at a distance D_(p) from thecenter of the closest resonant object, whose characteristic size isL_(R), and the ratio of the average electric field energy density to theaverage magnetic field energy density is smaller than about 0.1 forD_(p)/L_(R) less than
 3. 62. The apparatus of claim 57, wherein theresonant fields are primarily magnetic at a distance D_(p) from thecenter of the closest resonant object, whose characteristic size isL_(R), and the ratio of the average electric field energy density to theaverage magnetic field energy density is smaller than about 0.01 forD_(p)/L_(R) less than
 3. 63. The apparatus of claim 57, wherein at leastone of the first and second resonator structures comprises acapacitively loaded loop or coil of at least one of a conducting wire, aconducting Litz wire, and a conducting ribbon.
 64. The apparatus ofclaim 63, wherein the resonator structure receiving energy from theother resonator structure comprises a capacitively loaded loop of atleast one of a conducting wire, a conducting Litz wire, and a conductingribbon, wherein the characteristic size of said loop is less than 10 cmand the width of said conducting wire or Litz wire or ribbon is lessthan 1 cm.
 65. The apparatus of claim 63, wherein the resonatorstructure receiving energy from the other resonator structure comprisesa capacitively loaded loop of at least one of a conducting wire, aconducting Litz wire, and a conducting ribbon, wherein thecharacteristic size of said loop is less than 30 cm and the width ofsaid conducting wire or Litz wire or ribbon is less than 5 cm.
 66. Theapparatus of claim 57, wherein at least one of the first and secondresonator structures comprises an inductively loaded rod of at least oneof a conducting wire, a conducting Litz wire, and a conducting ribbon.67. The apparatus of claim 66, wherein the resonator structure receivingenergy from the other resonator structure comprises an inductivelyloaded rod of at least one of a conducting wire, a conducting Litz wire,and a conducting ribbon, wherein the characteristic size of said rod isless than 10 cm and the width of said conducting wire or Litz wire orribbon is less than 1 cm.
 68. The apparatus of claim 66, wherein theresonator structure receiving energy from the other resonator structurecomprises an inductively loaded rod of at least one of a conductingwire, a conducting Litz wire, and a conducting ribbon, wherein thecharacteristic size of said rod is less than 30 cm and the width of saidconducting wire or Litz wire or ribbon is less than 5 cm.
 69. Theapparatus of claim 38, further comprising a feedback mechanism formaintaining the resonant frequency of one or more of the resonantobjects.
 70. The apparatus of claim 69, wherein the feedback mechanismcomprises an oscillator with a fixed frequency and is configured toadjust the resonant frequency of the one or more resonant objects to beabout equal to the fixed frequency.
 71. The apparatus of claim 69, wherethe feedback mechanism is configured to monitor an efficiency of theenergy transfer, and adjust the resonant frequency of the one or moreresonant objects to maximize the efficiency.
 72. A method for wirelessenergy transfer, the method comprising: providing a first resonatorstructure and transferring energy with a second resonator structure,over a distance D larger than a characteristic size L₁ of said firstresonator structure and larger than a characteristic size L₂ of saidsecond resonator structure, wherein the energy transfer has a rate κ andis mediated by evanescent-tail coupling of a resonant field of the firstresonator structure and a resonant field of the second resonatorstructure, wherein said resonant field of the first resonator structurehas a resonance angular frequency ω₁, a resonance frequency-width Γ₁,and a resonance quality factor Q₁=ω₁/2Γ₁ at least larger than 300, andsaid resonant field of the second resonator structure has a resonanceangular frequency ω₂, a resonance frequency-width Γ₂, and a resonancequality factor Q₂=ω₂2/2Γ₂ at least larger than 300, wherein the absolutevalue of the difference of said angular frequencies ω₁ and ω₂ is smallerthan the broader of said resonant widths Γ₁ and Γ₂, and the quantityκ/√{square root over (Γ₁Γ₂)} is at least larger than
 20. 73. A methodfor wireless energy transfer, the method comprising: providing a firstresonator structure and transferring energy with a second resonatorstructure, over a distance D larger than a characteristic thickness T₁of said first resonator structure and larger than a characteristic sizeL₂ of said second resonator structure, wherein the energy transfer has arate κ and is mediated by evanescent-tail coupling of a resonant fieldof the first resonator structure and a resonant field of the secondresonator structure, wherein said resonant field of the first resonatorstructure has a resonance angular frequency ω₁, a resonancefrequency-width Γ₁, and a resonance quality factor Q₁=ω₁/2Γ₁ at leastlarger than 300, and said resonant field of the second resonatorstructure has a resonance angular frequency ω₂, a resonancefrequency-width Γ₂, and a resonance quality factor Q₂=ω₂/2Γ₂ at leastlarger than 300, wherein the absolute value of the difference of saidangular frequencies ω₁ and ω₂ is smaller than the broader of saidresonant widths Γ₁ and Γ₂, and the quantity κ/√{square root over (Γ₁Γ₂)}is at least larger than 20.